marshals, monotone marshals, and hypertree-width
TRANSCRIPT
Marshals, MonotoneMarshals, andHypertree-Width
Isolde AdlerDEPARTMENT OF MATHEMATICAL LOGIC
RM 205, UNIVERSITY OF FREIBURG
ECKERSTR. 1, D-79102 FREIBURG, GERMANY
E-mail: [email protected]
Received April 26, 2003; Revised February 23, 2004
Published online in Wiley InterScience(www.interscience.wiley.com).
DOI 10.1002/jgt.20025
Abstract: The tree-width of a hypergraph H equals one less than thenumber of cops necessary to catch the robber in theMonotone Robber andCops Game played on H. Analogously, the hypertree-width of a hypergraphis characterised by the Monotone Robber and Marshals Game. Whilethe Robber and Cops Game and its monotone variant coincide, Gottlob,Leone and Scarcello stated the corresponding question for the Robberand Marshals Game as an open problem. We give a negative answer.Concerning the generalised hypertree-width, our counterexamples showthat it is neither characterised by the Robber and Marshals Game nor byits monotone variant. We conclude by resuming how these hypergraphinvariants are related. � 2004 Wiley Periodicals, Inc. J Graph Theory 47: 275–296, 2004
Keywords: tree width; hypertree width; generalized hypertree width; hypergraph; game
� 2004 Wiley Periodicals, Inc.
275
1. INTRODUCTION
The notions of tree-width and hypertree-width are important devices for structural
investigations of graphs and hypergraphs. Many NP-complete problems become
polynomially solvable when restricted to hypergraphs H with bounded tree-width
or bounded hypertree-width (see [3,4]). There are classes of hypergraphs with
bounded hypertree-width, whose tree-width is unbounded, and where the tree-
width of the incidence graphs is also unbounded. In [6], Seymour and Thomas
introduce the Robber and Cops Game together with a monotone game variant,
both characterising the tree-width of a graph. In [5], Gottlob, Leone and Scarcello
show that the hypertree-width of a hypergraph is characterised by the monotone
variant of the Robber and Marshals Game. They state as an open problem,
whether the monotone variant can be replaced by the non-monotone variant.
In Section 2, the basic notions are introduced. In Sections 3 and 4, we give a
negative answer to the open problem by constructing a hypergraph where n
additional marshals are necessary for catching the robber monotonely. In Section 5,
we present a graph on which the games are non-equivalent. In Section 6, we use
this graph to investigate the relationship between the game invariants, hypertree-
width and generalised hypertree-width (introduced in [5]) of a hypergraph. We
express the generalised hypertree-width of a hypergraph H in terms of the
hypertree-width of the simplicialisation �H of H. This enables us to show that
the number of marshals necessary to catch the robber is a lower bound for
the generalised hypertree-width, and the number of marshals necessary to
catch the robber monotonely is an upper bound. But our examples show that the
generalised hypertree-width is neither characterised by the Robber and Marshals
Game nor by its monotone variant.
2. BASIC DEFINITIONS
For a set X; PðXÞ denotes the power set of X. P¼kðXÞ denotes the set of all
subsets of X with exactly k elements. An (undirected) graph is a pair G ¼ðVðGÞ;EðGÞÞ, consisting of a nonempty set VðGÞ of vertices and a set EðGÞ �P¼2ðVðGÞÞ of edges. A directed graph is a pair G ¼ ðVðGÞ;EðGÞÞ, where VðGÞis a nonempty set of vertices and EðGÞ � VðGÞ � VðGÞ is the set of edges.
Let G be a graph, X � VðGÞ. The graph hXi :¼ ðX;EðGÞ \ P¼2ðXÞÞ is called
the subgraph (of G) induced by X. For an integer n � 3, an n-cycle is a cycle C
with n vertices.1 A clique is a graph G with EðGÞ ¼ P¼2ðVðGÞÞ. A tree is a
connected, directed graph T containing no cycles, which satisfies:
(1) There is exactly one r 2 VðTÞ without any predecessors, the root of T .
(2) All vertices t 2 VðTÞ other than the root have exactly one predecessor,
denoted by predðtÞ.Let T be a tree and t 2 VðTÞ. Tt denotes the complete subtree of T with root t.
1For more detailed definitions the reader is referred to [2].
276 JOURNAL OF GRAPH THEORY
A hypergraph is a pair H ¼ ðVðHÞ;EðHÞÞ, consisting of a finite, nonempty set
VðHÞ of vertices, and a set EðHÞ � PðVðHÞÞ of hyperedges withSEðHÞ ¼
VðHÞ (see Fig. 1 for an example). Note that every graph is a hypergraph in a
natural way, provided that every vertex is contained in an edge. Let H be a
hypergraph and h 2 EðHÞ. We say h is an n-edge, if jhj ¼ n. Let X � VðHÞ. The
hypergraph H0 with VðH0Þ ¼ X and EðH0Þ ¼ fh \ X j h 2 EðHÞg is called the
subhypergraph (of H) induced by X. We denote it by H0 ¼ hXi. We write HnX for
hVðHÞnXi. Let n � 1 be an integer. A path (from v1 to vn) in a hypergraph H
is a sequence of vertices v1; . . . ; vn, s. t. two consecutive vertices are contained
in a common hyperedge of EðHÞ. Thus, the definitions of connectedness and
connected component can be transferred from graphs to hypergraphs in a natural
way.
All graphs and hypergraphs are finite.
A. Games
Let H be a hypergraph and k � 0 an integer. The Robber and k Marshals Game
on H, MarðH; kÞ, is played by two players, I and II, on the hypergraph H. Player I
plays k marshals and player II plays the robber. The marshals move on the
hyperedges of H, trying to catch the robber. In each move, some of the marshals
fly in helicopters to new hyperedges. The robber moves on the vertices of H.
He sees where the marshals will be landing and quickly tries to escape running
arbitrarily fast along paths of H, not being allowed to run through a marshal.
Player I’s objective is to land a marshal via helicopter on a hyperedge containing
the vertex occupied by the robber. Player II tries to elude capture. More precisely,
a position of MarðH; kÞ is a pair ðX; rÞ; where X 2 P�kðEðHÞÞ and r 2 VðHÞ.
FIGURE 1. A hypergraph H. Here, the 2-edges of H are depicted as graph edges.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 277
In the beginning of the game, player I chooses an X 2 P�kðEðHÞÞ. Player II
responds by choosing an r 2 VðHÞ. In each step of the game, say in position
ðX; rÞ, player I chooses an X0 2 P�kðEðHÞÞ and the marshals fly from X to X0.Player II chooses an r0 2 VðHÞ that is connected to r by a path in Hnð
SX \S
X0Þ. Player I wins, if a position ðX; rÞ with r 2SX is reached. Such a position
is called a capture position. Player II wins, if a capture position is never
reached.
Let H be a hypergraph and X � VðHÞ. A connected component C � VðHÞ of
HnX is called an escape space (with respect to X). It is easy to see that the
marshals’ moves need to depend only on the robber’s escape space (rather than on
the actual vertex he occupies).
Lemma 2.1. Let ðX; pÞ and ðX; qÞ be game positions with p 2 R and q 2 R,
where R is an escape space with respect to X. The marshals choose the new
position Y . Let ðY; rÞ be a position which the robber can reach from ðX; pÞ. Thenthe the robber can reach ðY ; rÞ from ðX; qÞ as well (and vice versa).
We use the previous lemma to represent a position which is not a capture
position by ðX;RÞ instead of ðX; pÞ, where R is the escape space with respect to Y
containing the robber. Let H be a hypergraph. For X;Y � EðHÞ and R � VðHÞwe call
RðY;X;RÞ :¼ fC � VðHÞ j C is a connected component of Hn[
Y s: t:
there is a path from R to C in Hnð[
X \[
YÞg
the set of possible escape spaces with respect to Y . So the set of possible escape
spaces with respect to Y is a subset of the set of all escape spaces with respect to
Y . It contains exactly those escape spaces with respect to Y which the robber can
in fact reach from his previous position. In the first move, RðY ; ;;VðHÞÞ ¼fR j R is an escape space with respect to Yg.
Let ðY ;RÞ be a game position. Player I makes a monotone move if she chooses
a Y 0 2 P�kðEðHÞÞ such that the robber’s new escape space R0 in every possible
move of player II is a proper subset of the previous escape space R: R0 �=R.
The Monotone Robber and k Marshals Game on H, Mon-MarðH; kÞ, is played
like MarðH; kÞ with the one difference that player I is only allowed to make
monotone moves. If there is a step in which player I cannot make a monotone
move, then player II wins. The marshal-width of H is
mwðHÞ :¼ minfk 2 N j player I has a winning strategy in MarðH; kÞg:
The monotone marshal-width of H is
mon-mwðHÞ :¼ minfk 2 N j player I has a winning strategy in
Mon-MarðH; kÞg:
278 JOURNAL OF GRAPH THEORY
For more convenience, we give a precise definition of a winning strategy.
Awinning strategy in MarðH; kÞ for the marshals is a triple ðT ; �; �Þ satisfying:
(a) T is a (finite, directed) tree.
(b) � : VðTÞ ! P�kðEðHÞÞ.(c) � : VðTÞ ! PðVðHÞÞ.(d) �ðrÞ ¼ VðHÞ, where r is the root of T, and
�ðsÞ 2 Rð�ðtÞ; �predðtÞ; �ðtÞÞ, if s 6¼ r and t ¼ predðsÞ.
Here, we used the abbreviation
�predðtÞ ¼; if t is the root of T ;�ðpredðtÞÞ otherwise:
�
(e) For each t 2 VðTÞ and each R 2 Rð�ðtÞ;�predðtÞ; �ðtÞÞ there is exactly one
successor s 2 VðTÞ with �ðsÞ ¼ R.
ðT; �; �Þ is a monotone winning strategy, if for all t 2 VðTÞ with t 6¼ r we have
�ðtÞ 6��ðpredðtÞÞ. Note that the marshals have a (monotone) winning strategy on a
hypergraph H if and only if they have a (monotone) winning strategy on every
connected component of H.
The Robber and k Cops Game on a hypergraph and its monotone variant
are defined analogously, with the only difference that the cops occupy vertices
instead of hyperedges2 and hence are less powerful than the marshals: modify
the previous definitions by replacing X 2 P�kðEðHÞÞ by X 2 P�kðVðHÞÞ,SX by
X; � : VðTÞ ! P�kðEðHÞÞ by � : VðTÞ ! P�kðVðHÞÞ andS�ðtÞ by �ðtÞ.
For the Robber and Cops Game on H the cop-width of H is
cwðHÞ :¼ minfk 2 N j k cops have a winning strategy in the Robber
and Cops Game}.
The monotone cop-width of H is
mon-cwðHÞ :¼ minfk 2 N j k cops have a monotone winning strategy in
the Robber and Cops Gameg:
B. Decompositions
Let H be a hypergraph. A hypertree-decomposition of H is a triple ðT ; �; �Þ,consisting of a tree T and two labelling functions � : VðTÞ ! PðVðHÞÞ and
� : VðTÞ ! PðEðHÞÞ satisfying:
2See [6]. Note that the definition of the Robber and Cops Game for graphs in [6] can be translated literally from
graphs to hypergraphs by replacing the word ‘graph’ by ‘hypergraph’ and ‘edge’ by ‘hyperedge’, yielding our
notion.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 279
(1) For each hyperedge h 2 EðHÞ there is a vertex t 2 VðTÞ such that h � �ðtÞ (t
covers h).
(2) For each vertex v 2 VðHÞ the set ft 2 VðTÞ j v 2 �ðtÞg is connected.
(3) Each node t 2 VðTÞ satisfies �ðtÞ �S�ðtÞ.
(4) Each node t 2 VðTÞ satisfies �ðTtÞ \S�ðtÞ � �ðtÞ, where �ðTtÞ ¼S
s2VðTtÞ �ðsÞ.
The width of a hypertree-decomposition of H is defined as
wðT ; �; �Þ :¼ maxf �ðtÞj j j t 2 VðTÞg:
The hypertree-width of H is defined as
htwðHÞ :¼ minfwðT; �; �Þ j ðT; �; �Þ hypertree-decomposition of Hg:
Figure 2 shows a hypertree-decomposition of width 2 of the hypergraph H
from Figure 1. The rectangles are the tree nodes. The left part of a rectangle
represents the � label and the right part represents the � label.
A pair ðT; �Þ satisfying (1) and (2) is called a tree-decomposition of H.
The width of a tree-decomposition of H is defined as
wðT ; �Þ :¼ maxf �ðtÞj j � 1 j t 2 VðTÞg:
The tree-width of H is defined as
twðHÞ :¼ minfwðT; �Þ j ðT; �Þ tree-decomposition of Hg:
The definitions of tree-decomposition and tree-width from [6] can be translated
literally from graphs to hypergraphs, then coinciding with our definitions. Note
that the hypertree-width (tree-width) of a hypergraph H equals the maximum of
FIGURE 2. A width 2 hypertree-decomposition of the hypergraph from Figure 1.
280 JOURNAL OF GRAPH THEORY
the hypertree-widths (tree-widths) of its connected components. Now we can
formulate the game theoretic characterisations as follows:
Lemma 2.2. (Gottlob, Leone and Scarcello [5]).Let H be a hypergraph. Then, mon-mwðHÞ ¼ htwðHÞ.Lemma 2.3. (Seymour and Thomas [6]).Let H be a hypergraph. Then, mon-cwðHÞ ¼ twðHÞ þ 1.
3. MONOTONICITY COST
Theorem 3.1 (Seymour and Thomas [6]). Let H be a hypergraph. If k cops have
a winning strategy on H, then k cops have a monotone winning strategy on H.
Thus, mon-cwðHÞ ¼ cwðHÞ.In this section, we show that this result cannot be transferred to the Robber
and Marshals Game. More precisely, we will construct a hypergraph H1 with
mon-mwðH1Þ ¼ 4 and mwðH1Þ ¼ 3, that is a hypergraph with monotonicity cost
mon-mwðHÞ � mwðHÞ ¼ 1. Note that mon-mwðHÞ � mwðHÞ � 0 for every
hypergraph H. The following easy remark will be used tacitly.
Remark 1. An n-cycle C satisfies mon-mwðCÞ ¼ mwðCÞ ¼ 2.
Let H be a hypergraph, v;w 2 VðHÞ, v 6¼ w. Connecting v and w with a 2-edge
means passing from H to the hypergraph H0 with H0 ¼ ðVðHÞ;EðHÞ [ ffv;wggÞ.Example 1. Let H1 be the following hypergraph (see Fig. 3).
Let B :¼ fgij j i; j 2 f1; 2gg [ fg0ij j i; j 2 f1; 2gg,
VðH1Þ :¼ B [ f0; 1; 2; 3; 4; 00; 10; 20; 30; 40g and
EðH1Þ :¼ ffp; gg j p 2 VðH1ÞnB and g 2 Bg [ fa1; a2; b1; b2g [ ff40; 20g;f20; 30gg[
ff20; 10g; f10; 00g; f30; 00g; f00; 0g; f3; 0g; f1; 0g; f2; 1g; f2; 3g; f4; 2gg,
where a1 ¼ fg11; g12; g011; g
012; 4
0g; a2 ¼ fg21; g22; g021; g
022; 3g,
b1 ¼ fg11; g21; g011; g
021; 4g and b2 ¼ fg12; g
012; g22; g
022; 3
0g.
The vertices from B will be called balloon vertices, the others will be called
ground vertices.
This example is based on the idea of forcing the marshals to occupy a certain
vertex (4 or 40) too early. This vertex will be added to the robber’s escape space
again before he can be caught.
Figure 4 shows a monotone winning strategy for four marshals and a non-
monotone winning strategy for three marshals on H1, where every edge
ðpredðtÞ; tÞ of a strategy tree t represents a possible escape space �ðtÞ, indicated
by one of its elements. Every vertex t is a position 6ðtÞ of the marshals. It remains
to show that there is no monotone winning strategy for less than four marshals on
H1 and that there is no winning strategy for less than three marshals on H1 at all.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 281
Claim 3.1. Let X 2 P�3ðEðH1ÞÞ be a game position on H1 of �3 marshals.
(a) VðH1ÞnSX contains at least one ground vertex.
(b) IfSX avoids a balloon vertex, that is B 6�
SX, then VðH1Þn
SX is
connected.
Proof. (a) There are 10 ground vertices and 4 marshals can occupy at most 8
of them at the same time.
FIGURE 4. A monotone winning strategy for four marshals (left) and a non-monotone winning strategy for three marshals on H 1 (right). Every edge of thestrategy tree represents a possible escape space, indicated by one of itselements. The non-monotone move is the move where the marshals occupy b1and b2 for the first time.
FIGURE 3. H1: additionally, every balloon vertex is connected to every groundvertex by a 2-edge. These edges are indicated by coronas around the balloonvertices.
282 JOURNAL OF GRAPH THEORY
(b) Let b 2 BnSX. Since b is a balloon vertex, b is connected to every ground
vertex from VðH1ÞnSX. By (a), VðH1Þn
SX contains a ground vertex g, hence b
is also connected (via g) to any remaining vertex from VðH1ÞnSX. &
Claim 3.2. Let ðT ; �; �Þ be a winning strategy for H1 for � 3 marshals. Suppose
ðT; �; �Þ is minimal in the sense that no proper subtree Tt of T induces a winning
strategy ðTt; �jTt ; �jTtÞ for H1. Then, B �S�ðtÞ holds for all t 2 VðTÞ. Hence,
fa1; a2g � �ðtÞ or fb1; b2g � �ðtÞ for each t 2 VðTÞ.
Proof. Suppose t 2 VðTÞ, but BnS�ðtÞ 6¼ ;. By Claim 3.1, the subhyper-
graph VðH1ÞnSX is connected, that is t has only one successor s, and the subtree
Ts yields a winning strategy ðTs; �jTs ; �jTsÞ, a contradiction. &
Claim 3.3. If B �SX and X 2 P�3ðEðH1ÞÞ, then fa1; a2g � X or fb1; b2g �
X.
Proof. Suppose fa1; a2g 6� X and fb1; b2g 6� X. Then X contains zero, one or
two edges from fa1; a2; b1; b2g, and these cover zero, four or six vertices from B,
respectively. Each of the remaining three, two or one edges cover at most one
vertex from B. Thus X covers at most seven vertices from B, a contradiction.
&
Claim 3.4. There is no winning strategy of width < 3 for H1.
Proof. Otherwise, there is a minimal winning strategy ðT; �; �Þ of width 2.
By Claim 3.2 the two marshals occupy B in each node of the strategy. Hence, they
occupy a1 and a2 or b1 and b2, respectively. Meanwhile, the unmolested robber
remains comfortably seated on vertex 1, watching the marshals‘win’. &
Claim 3.5. There is no monotone winning strategy of width 3 for H1.
Proof. Otherwise, there is a minimal monotone winning strategy ðT; �; �Þ of
width 3. Claim 3.2 implies B �S�ðtÞ for all t 2 VðTÞ. By Claim 3.3, for every
t 2 VðTÞ either fa1; a2g � �ðTÞ or fb1; b2g � �ðTÞ. We now show that the
robber can always escape. Let C be the cycle with vertices f0; 1; 2; 3g and let C0
be the cycle with vertices f00; 10; 20; 30g. Let r be the root of T . Without restriction
fa1; a2g � �ðrÞ. Hence, player I begins by placing two marshals (the switch
marshals) on a1 and a2, and the third marshal on an arbitrary hyperedge. Player II
responds by placing the robber on a vertex in the cycle C0. This is possible,
because at least one vertex of C0 is not occupied by the marshals. Since
mon-mwðC0Þ ¼ 2, the third marshal alone cannot expel the robber from C0. The
only possible move for player I is to place two marshals on b1 and b2, which
contradicts monotonicity, because the escape space is extended either by vertex 3
or by vertex 40. &
Putting all things together, we have mon-mwðH1Þ ¼ 4 and mwðH1Þ ¼ 3.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 283
4. HIGHER MONOTONICITY COST
Theorem 4.1. For each n 2 N there exists a hypergraph Hn with monotonicity
cost n, that is mon-mwðHnÞ ¼ mwðHnÞ þ n.
H1 from Example 1 shows this for n ¼ 1. For arbitrary n 2 N we will now
construct an example Hn, proving the theorem. This is done by appropriately
generalising the construction that forced the marshals to make a non-monotone
move in the three-marshal winning strategy on H1. On H1 there were exactly two
possibilities for two marshals to occupy all the balloon vertices. They could either
occupy the hyperedges a1 and a2, or b1 and b2. Thus, intuitively we had exactly
two ‘switch settings’. In one switch setting, the vertices 3 and 40 had necessarily
to be occupied as well, in the other the vertices 30 and 4 had to be occupied as
well.
Let n � 0 be an integer. A switch graph (with n-switch) is a triple ðH; �; �Þ,where H is a hypergraph and � ¼ f�1; . . . ; �ng; � ¼ f�1; . . . ; �ng are subsets of
EðHÞ with j�j ¼ j�j ¼ n.
The n specified hyperedges �1; . . . ; �n are called the � switch edges,
�1; . . . ; �n are called the � switch edges.
Now, we define a game and its monotone variant suitable for switch graphs.
Subsequently, we will construct a switch graph with monotonicity cost n.
Theorem 4.2 shows that a switch graph with monotonicity cost n for the switch
games yields a hypergraph with monotonicity cost n for the Robber and Marshal
Games.
Let ðH; �; �Þ be a switch graph with n-switch, n � k. In the switch game on
ðH; �; �Þ with k marshals, SwitchðH; �; �; kÞ, player I plays k marshals. In each
move, n fixed marshals, the switch marshals, have to be in position � or �, that is
they have to occupy all the � switch edges or all the � switch edges. The other
marshals and the robber act as in the Robber and Marshals Game. In the
monotone switch game on ðH; �; �Þ, Mon-SwitchðH; �; �; kÞ, only monotone
moves are allowed.
The marshal-width of the switch graph ðH; �; �Þ is defined as
smwðH; �; �Þ :¼ minfk 2 N j k marshals have a winning strategy in
the switch game on ðH; �; �Þg:
The monotone marshal-width of the switch graph ðH; �; �Þ is defined as
mon-smwðH; �; �Þ :¼ minfk 2 N j k marshals have a monotone winning
strategy in the switch game on ðH; �; �Þg:
Before constructing the switch graph with monotonicity cost n, we need some
definitions and a lemma.
284 JOURNAL OF GRAPH THEORY
A ðk; k0Þ-hypergraph is a hypergraph H with mwðHÞ ¼ k and
mon-mwðHÞ ¼ k0. A ðk; k0Þ-switch graph is a switch graph ðH; �; �Þ with
smwðH; �; �Þ ¼ k and mon-smwðH; �; �Þ ¼ k0.
Lemma 4.1. Let H be a ðk; kÞ-hypergraph. Then H0 ¼ ðVðHÞ;EðHÞ [ffvg j v 2 VðHÞgÞ is a ðk; kÞ-hypergraph as well.
Proof. Let mon-mwðHÞ ¼ mwðHÞ ¼ k. Since H0 has the same paths as H,
the connected subsets are identical. Hence, every winning strategy ðT ; �; �Þ for
Mon-MarðH; kÞ is also a winning strategy for Mon-MarðH0; kÞ, because the
possible escape spaces are the same. Therefore, mwðH0Þ � mon-mwðH0Þ � k.
On the other hand mwðH0Þ � k, since a winning strategy for MarðH0; k � 1Þwould yield one for MarðH; k � 1Þ. &
Hence, adding for every vertex v of a ðk; kÞ-hypergraph the hyperedge fvg is
no advantage to the marshals.
A punctured hypergraph is a hypergraph H with ffvg j v 2 VðHÞg � EðHÞ.Let H be a hypergraph, w =2 VðHÞ. The cone with apex w over H is the
hypergraph H0, where VðH0Þ ¼ VðHÞ [_ fwg and EðH0Þ ¼ EðHÞ [ ffv;wg jv 2 VðHÞg. Note that for every integer n � 0, there exists a hypergraph N with
mwðNÞ ¼ mon-mwðNÞ > n: an m� m grid for suitable m, for example. The
following switch-graph has monotonicity cost n.
Example 2. Let n � 2. Let N1 be a punctured hypergraph with mwðN1Þ ¼mon-mwðN1Þ > n. Let N2 be a disjoint copy of N1. Let ðH; �; �Þ be the switch
graph with
VðHÞ : ¼ VðN1Þ [_ VðN2Þ [_ fmg [_ fe1i j i ¼ 1; . . . ; nþ 1g[_ fe2i j i ¼ 1; . . . ; nþ 1g:
The vertices m; e11; . . . ; e1;nþ1 are apices of cones over N1 and the vertices
m; e21; . . . ; e2;nþ1 are apices of cones over N2. We set
EðHÞ : ¼ EðN1Þ [_ EðN2Þ[_ ffe1ig j i ¼ 1; . . . ; nþ 1g [_ ffe2ig j i ¼ 1; . . . ; nþ 1g [_ ffmgg[_ ffe1i; n1g j i ¼ 1; . . . ; nþ 1; n1 2 VðN1Þg[_ ffe2i; n2g j i ¼ 1; . . . ; nþ 1; n2 2 VðN2Þg[_ ffm; ng j n 2 VðN1Þ [ VðN2Þg
and
� ¼ ffe11g; . . . ; fe1;nþ1g; fmgg [ VðN2Þ,� ¼ ffe21g; . . . ; fe2;nþ1g; fmgg [ VðN1Þ.
Each 1-edge from N1 and each apex over a cone of N2 is an � switch edge, and
each 1-edge from N2 and each apex over a cone of N1 is a � switch edge.
Figure 5 shows this switch graph.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 285
Claim 4.1. ðH; �; �Þ is a ðk; k0Þ-switch graph with monotonicity cost k0 � k ¼ n
for the switch games.
Proof. Let s :¼ jN1j þ ðnþ 1Þ þ 1 be the number of switch marshals, that is
s ¼ j�j ¼ j�j. Figure 6 shows a (non-monotone) winning strategy for the switch
game with sþ 1 marshals and Figure 7 shows a monotone winning strategy for
the switch game with sþ ðnþ 1Þ marshals. There is no winning strategy for the
switch game with s marshals, since the robber can escape as follows: when the
marshals are in position �, he stays anywhere in N1 and when they are in position
�, he goes to e11. Hence, smwðH; �; �Þ ¼ sþ 1.
There is no monotone winning strategy for the switch game with sþ n
marshals. We may assume that at first the switch marshals are in position �.
The robber can escape by moving to N1. By choice of N, n marshals cannot catch
him there. Since n marshals cannot occupy e11; . . . ; e1;nþ1 at the same time, the
marshals cannot switch to position � without making a non-monotone move.
Hence, mon-smwðH; �; �Þ ¼ sþ nþ 1 and mon-smwðH; �; �Þ � smwðH;�; �Þ ¼ n. &
A. Implementing Switch Graphs as Hypergraphs
Now, we will point out how a ðk; k0Þ-switch graph ðH; �; �Þ can be transformed
into a ðk; k0Þ-hypergraph �ðH; �; �Þ with the same marshal-widths, by gluing to it
a suitable hypergraph.
FIGURE 5. The switch graph (H,�,�) with monotonicity cost n.
FIGURE 6. A non-monotone winning strategy for the switch game on the switchgraph from Figure 5 with sþ 1 marshals.
286 JOURNAL OF GRAPH THEORY
Let H be a ðk; k0Þ-hypergraph. Then, MarðH; kÞ (Mon-MarðH; k0Þ) is called the
minimal safe (monotone) game on H. Let ðH; �; �Þ be a ðk; k0Þ-switch graph.
SwitchðH; �; �; kÞ (Mon-SwitchðH; �; �; kÞ) is called the minimal safe (mono-
tone) game on ðH; �; �Þ.For each winning strategy in the minimal safe (monotone) game
Marð�ðH; �; �Þ; kÞ (and Mon-Marð�ðH; �; �Þ; k0Þ), the occupation of all edges
from � or all edges from � will be necessary in every move except in the capture
positions. As we will see, the winning strategies for SwitchððH; �; �Þ; kÞ ‘are’ the
winning strategies for Marð�ðH; �; �Þ; kÞ, and the same holds for the monotone
variants.
Let H be a hypergraph. We call two hyperedges h and h0 2 EðHÞ anchored, if
there exists a vertex v 2 VðHÞ (called eyelet vertex) satisfying v =2 h; v =2 h0;h [ fvg; h0 [ fvg 2 EðHÞ. Here, we allow3 fvg 2 EðHÞ, but apart from that, the
two hyperedges h [ fvg and h0 [ fvg (called the anchor edges) are the only
further hyperedges from H containing v. h and h0 are anchored m times with
eyelet vertices v1; . . . ; vm (pairwise distinct), if for each of the vertices vi;i ¼ 1 . . . n; h and h0 are anchored with eyelet vertex vi. A BOG-hypergraph (of
order m) is a hypergraph HBOG together with a partitioning VðHBOGÞ ¼ B _[[O _[[Gof VðHBOGÞ, satisfying: For all b 2 B and g 2 G, there are hyperedges
hb; hg 2 EðHBOGÞ, s. t. b 2 hb � B, g 2 hg � G and hb and hg are anchored m
times with eyelet vertices v1; . . . vm 2 O. We call the vertices from G ground
vertices, the vertices from B balloon vertices, and the vertices from O eyelet
vertices.
Let X � VðHÞ. The covering number of X is
�ðXÞ :¼ minfn 2 N j n marshals can cover all vertices from Xg:
In a game played on a BOG-hypergraph with large order m the marshals either
cover all vertices from G and play on hBi, or they cover all vertices from B and
play on hGi. In both cases, the role of the eyelet vertices is of little significance,
they only make sure that a marshal, when occupying a balloon vertex, cannot
interfere in the game played on the ground vertices by occupying a ground vertex
3For the construction, it is irrelevant whether fvg 2EðH Þ or not. In Section 5, we will prove the existence of
simplical hypergraphs with monotonicity cost. For that purpose we allow fvg 2EðH Þ.
FIGURE 7. A monotone winning strategy for s þ (n þ 1) marshals for the switchgame on (H,�,�) from Figure 5.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 287
at the same time, while at the same time, less than m marshals cannot break the
connection between any uncovered vertex of B and any uncovered vertex of G.
More precisely:
Lemma 4.2. Let HBOG be a BOG-hypergraph of order m > mon-mwðhGiÞþ�ðBÞ. Moreover, suppose �ðGÞ > mon-mwðhGiÞ þ �ðBÞ. The following state-
ments hold.
(a) mwðHBOGÞ � mwðhGiÞ þ �ðBÞ, and mon-mwðHBOGÞ � mon-mwðhGiÞþ�ðBÞ.
(b) Let ðT ; �; �Þ be a winning strategy for at most mon-mwðHBOGÞ marshals.Then G 6�
S�ðtÞ for all t 2 VðTÞ.
(c) Under the assumptions of (b), all nodes t 2 VðTÞ satisfy:If �ðtÞ \ ðB [ GÞ 6¼ ; and B 6�
S�ðtÞ, then t has a successor s such that
�ðsÞ � ðB [ GÞnS�ðtÞ 6¼ ;.
(d) There is a winning strategy ðT ; �; �Þ for the minimal safe game on HBOG,
satisfying for all t 2 VðTÞ: t is a leaf or B �S�ðtÞ.
The same holds for the minimal safe monotone game.
(b) States that the marshals cannot cover the ground vertices completely.
(c) Expresses: if the robber’s escape space contains a vertex of B [ G and in
the next move the marshals do not cover B completely, then the robber can reach
every unoccupied vertex of B [ G during the flight.
Proof of the lemma. (a) mwðHBOGÞ � mwðhGiÞ þ �ðBÞ, because �ðBÞ marsh-
als can cover the balloon vertices permanently, implying that the actual game
with the remaining marshals only takes place in hO [ Gi: there they play
according to their strategy for hGi, until they would have caught the robber on
hGi. At this point, the escape space consists of a single vertex v 2 O. By moving
to fhb [ fvg; hgg, where hb; hg are the hyperedges anchored via v, the marshals
can now win. Observe that monotonicity of a winning strategy is preserved by
this translation from hGi to HBOG.
(b) By assumption, we do not have enough marshals to cover B.
(c) Since B 6�S�ðtÞ there is a vertex b 2 Bn
S�ðtÞ. Since, mwðHBOGÞ �
mon-mwðHBOGÞ < �ðGÞ, by (b) there is a vertex g 2 GnS�ðtÞ. By (a)
m > mon-mwðHBOGÞ � mwðHBOGÞ, hence every b0 2 BnS�ðtÞ is connected to
g in HBOGnS�ðtÞ. Symmetrically, every g0 2 Gn
S�ðtÞ is connected to b in
HBOGnS�ðtÞ.
Therefore, there is an escape space R � ðB [ GÞnS�ðtÞ 6¼ ; with respect to
�ðtÞ. Let v 2 �ðtÞ \ ðB [ GÞ. Since v is connected to b or g during the flight, R is
a possible escape space.
(d) Let ðT ; �; �Þ be a winning strategy for the minimal safe (monotone) game.
Claim 4.2. We may assume that for each t 2 VðTÞ; �ðtÞ \ ðB [ GÞ 6¼ ; or t is a
leaf.
288 JOURNAL OF GRAPH THEORY
Proof of the Claim. Suppose �ðtÞ \ ðB [ GÞ ¼ ;. Then �ðtÞ ¼ fvg, for a
v 2 O since every neighbour of a vertex v 2 O is in B or in G. Therefore, we
may assume that �ðtÞ ¼ fhb [ fvg; hgg where hb and hg are anchored via v (the
marshals can be expected to finish the game when they can), and hence t is a leaf,
which proves the claim.
Now suppose, there is a t 2 VðTÞ s. t. B 6�S�ðtÞ and t is not a leaf. (*)
We may assume that t has maximum distance from the root of T . Then
�ðtÞ \ ðB [ GÞ 6¼ ; by the claim. By (c) there is a successor s of t s. t.
�ðsÞ � ðB [ GÞnS�ðtÞ 6¼ ;.
Define ðT 0; �0; �0Þ as follows:
T 0 ¼ Ts [ fone successor sv of s for each escape space fvg; v 2 OnS�ðsÞ,
for which there is no successor of s in Tsg,
�0ðt0Þ ¼ �ðt0Þ for t0 2 VðTsÞ;fhb [ fvg; hgg for t0 ¼ sv ðwhere hb and hg are anchored via vÞ;
�
�0ðt0Þ ¼ �ðt0Þ for t0 2 VðTsÞ;fvg for t0 ¼ sv:
�
Clearly ðT 0; �0; �0Þ is again a (monotone) winning strategy and t0 contains no
node satisfying ð�Þ. &
Let ðH; �; �Þ be a switch graph with n-switch, where � ¼ f�1; . . . ; �ng;� ¼ f�1; . . . ; �ng. The hypergraph with n-mechanics associated with ðH; �; �Þis the following BOG-hypergraph HBOG: G :¼ VðHÞ is the set of ground
vertices of HBOG and B :¼ fgij j i; j ¼ 1; . . . ng is the set of balloon vertices
(with n2 elements). Let m :¼ mon-mwðhGiÞ þ nþ 1 be the order of HBOG
and let O :¼ fplðg; hÞ j g 2 B; h 2 EðHÞ; l 2 f1; . . .mgg. Moreover, let EB :¼ffgijg j i; j ¼ 1; . . . ng [ fa1; . . . ; ang [ fb1; . . . ; bng, with ai :¼ fgi1; . . . ; ging[�i and bj :¼ fg1j; . . . ; gnjg [ �j; i; j ¼ 1; . . . ; n. Let EðHBOGÞ :¼ EB [ EðHÞ[S
g2B;h2EðHÞ;l2f1;...;mgffgg [ fplðg; hÞg; fplðg; hÞg [ hg.
Note that by definition n ¼ �ðBÞ and hence, m ¼ mon-mwðhGiÞþnþ 1 > mon-mwðhGiÞ þ �ðBÞ.
Let HBOG be the BOG-hypergraph with n-mechanics associated with ðH; �; �Þ.The marshals operate the n-mechanics in (Mon-)MarðHBOGÞ, if in every move
they occupy the hyperedges a1; . . . ; an or the hyperedges b1; . . . ; bn.We only admit ðk; k0Þ-switch graphs ðH; �; �Þ satisfying mon-mwðHÞ þ j�j <
�ðGÞ and j�j þ 1 � k. Now we will show, if ðH; �; �Þ is such a ðk; k0Þ-switch
graph, then the hypergraph with n-mechanics HBOG is a ðk; k0Þ-hypergraph. Let �be the map:
� :ðk; k0Þ-switch graphs
with witnesses
ðT; �; �Þ and ðT 0; �0; �0Þ
8<:
9=; !
ðk; k0Þ-hypergraphs with
mechanics and witnesses
ðT; �; �Þ and ðT 0; �0; �0Þ
8<:
9=;
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 289
ðH; �; �Þ; ðT ; �; �Þ; ðT 0; �0; �0Þ 7! �ðH; �; �Þ; �ðT ; �; �Þ; �ðT 0; �0; �0Þ,where �ðH; �; �Þ ¼ HBOG is the hypergraph with j�j-mechanics associated with
ðH; �; �Þ, and � transforms the strategy ðT; �; �Þ (and the strategy ðT 0; �0; �0Þ as
well) as follows.
In HBOG, the switch marshals operate the mechanics instead of occupying the
hyperedges a1; . . . ; an: In the strategy ðT; �; �Þ they occupy the switch edges
�1; . . . ; �n if and only if in �ðT ; �; �Þ they occupy the hyperedges a1; . . . ; an, and
equally for the bi and �i. Apart from that, the marshals play according to the
strategy ðT ; �; �Þ. Here, the escape spaces may contain additional eyelet vertices.
If the escape space contains other vertices next to the eyelet vertices, the
marshals move according to ðT; �; �Þ, as if the eyelet vertices did not exist. If the
escape space consists of a unique eyelet vertex, the marshals catch the robber in
their next move. Since j�j < k, this is always possible. We proceed equally with
ðT 0; �0; �0Þ. From Lemma 4.2 we infer:
Remark 1. If HBOG ¼ �ðH; �; �Þ and n ¼ j�j, then we may assume that the
marshals operate the n-mechanics in every winning strategy for the minimal safe
game on HBOG (except from the capture positions).
Altogether we have:
Theorem 4.2. Let ðH; �; �Þ be a ðk; k0Þ-switch graph with witnesses ðT ; �; �Þand ðT 0; �0; �0Þ; j�j ¼ j�j ¼ n. Then, the BOG-hypergraph HBOG associated with
ðH; �; �Þ is a ðk; k0Þ-hypergraph with n-mechanics, witnessed by �ðT ; �; �Þ and
�ðT 0; �0; �0Þ.
Hence, we obtain a hypergraph Hn with monotonicity cost n from the switch
graph ðH; �; �Þ from Example 2, by setting Hn :¼ HBOG, where HBOG is the
BOG-hypergraph with n-mechanics associated with ðH; �; �Þ. This completes the
proof of Theorem 4.1. &
5. MONOTONICITY COST ON GRAPHS
Since all our previous examples contain ‘big’ hyperedges, especially hyperedges
h with jhj > 2, the question arises, whether these hyperedges are necessary for
producing monotonicity cost. In this section, we give a negative answer by pre-
senting a graph with monotonicity cost 1.
Recall that the construction of the Examples 1 and 2 is based on the idea of
forcing the marshals (via balloon vertices) to occupy certain vertices ‘too early’
(e.g., the vertices 4 or 40 in H1 or the switch edges of Hn).
We call a hypergraph H simplicial, if PðhÞ � EðHÞ for all h 2 EðHÞ. By �Hwe denote the simplicialised hypergraph, where Vð�HÞ ¼ VðHÞ and Eð�HÞ ¼S
h2EðHÞ PðhÞ. On �H1 and on �Hn the marshals can avoid occupying those
vertices too early, so both hypergraphs have monotonicity cost 0. Do simplicial
hypergraphs have monotonicity cost 0? We will now see that the answer is no.
290 JOURNAL OF GRAPH THEORY
Remark 2. Let H be a hypergraph. If �H has monotonicity cost � 1, then H
has monotonicity cost � 1.
Proof. Note that mwðHÞ¼ mwð�HÞ and mon-mwðHÞ� mon-mwð�HÞ. &
In this section, we briefly present a simplicialised graph with monotonicity cost 1.
Hence by Remark 2, we also obtain a graph with monotonicity cost � 1. This
graph is also of interest when considering generalised hypertree-width, as we will
see in Section 6.
Example 3. Let H0 be the hypergraph with
VðH0Þ ¼ fa; b1; b2; c; d1; d2; e; f1; f2; g; h1; h2; ig and
EðH0Þ ¼S
j¼1;2ffa; ig; fa; bjg; fbj; cg; fc; djg; fdj; eg; fe; fjg; ffj; gg; fg; hjg;fhj; igg.
It is straightforward to show that mwðH0Þ ¼ mon-mwðH0Þ ¼ 3.
H0 helps us to construct the example.
Example 4. The hypergraph G1BOG is the following hypergraph (Fig. 8): let
B :¼ fg1; g2; g3; g4g be the set of balloon vertices with B \ VðH0Þ ¼ ;, where H0
is the hypergraph from Example 3. Let G :¼ VðH0Þ be the set of ground vertices,
ffgg j g 2 Gg � EðG1BOGÞ and ffbg j b 2 Bg � EðG1
BOGÞ.In addition, each hyperedge fgg; g 2 G, is anchored six times with each
hyperedge fbg; b 2 B. Call the set of necessary eyelet vertices O and let
ffpg j p 2 Og � EðG1BOGÞ. We set VðG1
BOGÞ :¼ G _[[O _[[B and
EðG1BOGÞ :¼ EðH0Þ [ ffa; g1g; fg1; g2g; fg2; f1g; fd2; g3g; fg3; g4g; fg4; igg[
ffpg j p 2 VðG1BOGÞg.
Note that G1BOG is simplicial.
Claim 5.1. G1BOG satisfies mon-mwðG1
BOGÞ ¼ 5 and mwðG1BOGÞ ¼ 4.
FIGURE 8. G1BOG from Example 4, for the sake of clearness, the eyelet vertices
are left out. Each of the hyperedges {g1} ,. . ., {g4} is anchored six times with eachhyperedge {g}, g2G.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 291
Proof. Figure 9 shows a (non-monotone) winning strategy for four marshals.
For the sake of clearness, the game moves necessary for the eyelet vertices are
left out. They do not affect the number of necessary marshals anyway. A
monotone winning strategy for five marshals can be obtained as follows: during
the whole game, two marshals occupy the hyperedges fg1; g2g; fg3; g4g. Apart
from the eyelet vertices, the unoccupied part of the graph is �H0. There, the
robber is easily caught by three further marshals.
There is no winning strategy for three marshals at all: obviously, �ðBÞ ¼ 2,
where B is the set of balloon vertices from G1BOG. It is easy to see that
mon-mwðhGiÞ ¼ 3 (where G is the set of ground vertices from G1BOG). Thus, by
Lemma 4.2 we may assume that the marshals occupy B in every winning strategy
for the minimal safe game (except for the capture positions). It is easy to see that,
if there is a winning strategy for three marshals on G1BOG, then there is a winning
strategy on hGi for two marshals, a contradiction.
It remains to prove, that there is no monotone winning strategy for four
marshals. This can be shown by considering all possible game evolutions with
four marshals backwards, that is beginning from the last move in a game and then
considering all possible previous positions. This is rather elaborate and will be
skipped here.4
4See [1] for details.
FIGURE 9. A (non-monotone) winning strategy for four marshals on G1BOG from
Example 4.
292 JOURNAL OF GRAPH THEORY
Let G1 be the graph obtained from G1BOG by removing all 1-edges. With
Remark 2, it is easy to see that G1 has monotonicity cost exactly 1.
6. GENERALIZED HYPERTREE-WIDTH
Dropping condition (4) from the definition of hypertree-decomposition, we obtain
the notion of generalised hypertree-decomposition (introduced in [5]). Corre-
spondingly, we obtain the generalised hypertree-width of H; ghtwðHÞ. Note that
every hypertree-decomposition is a generalised hypertree decomposition.
Lemma 6.1. Let H be a hypergraph. Then, ghtwðHÞ ¼ htwð�HÞ.
Proof. Let ðT ; �; �Þ be a generalised hypertree-decomposition of H of width
k. For t 2 VðTÞ we set �0ðtÞ :¼ fh \ �ðtÞ j h 2 �ðtÞg. It is easy to see that
ðT; �; �0Þ is a hypertree-decomposition of width � k of �H. Hence, htwð�HÞ �ghtwðHÞ. Conversely, let ðT ; �; �Þ be a hypertree-decomposition of width k of
�H. For each hyperedge h 2 �H let ’ðhÞ be a hyperedge of H with h � ’ðhÞ.For t 2 VðTÞ let �0ðtÞ :¼ f’ðhÞ j h 2 �ðtÞg. It follows immediately that ðT ; �; �0Þis a generalised hypertree-decomposition of width � k of H. &
Does every generalised hypertree-decomposition of width k induce a
hypertree-decomposition of width k? As we will see, the answer is no.
Claim 6.1. H1 from Example 1 satisfies, htwðH1Þ ¼ 4 and ghtwðH1Þ ¼ 3.
Proof. By Lemma 2.2, htwðH1Þ ¼ mon-mwðH1Þ ¼ 4. A generalised hyper-
tree-decomposition of width 3 can be constructed easily by modifying the
winning strategy for three marshals on H1 (see Fig. 4). On the other hand, if there
were a generalised hypertree-decomposition of width �2, by Lemma 6.1 two
marshals would have a winning strategy on �H1. It is easy to see that this is not
the case. &
Note that by Lemma 6.1, ghtwðHÞ ¼ htwð�HÞ ¼ htwð��HÞ ¼ ghtwð�HÞholds for all hypergraphs H. If H is simplicial, then ghtwðHÞ ¼ htwðHÞ. This
shows:
Claim 6.2. G1BOG from Example 4 satisfies, ghtwðG1
BOGÞ ¼ 5 and mwðG1BOGÞ ¼
4. &
Hence, the (non-monotone) marshal-width differs from the generalised hypertree-
width in general.
Corollary 6.1. Let H be a hypergraph. Then, mwðHÞ � ghtwðHÞ.
Proof. It is easy to see that mwðHÞ ¼ mwð�HÞ. Hence, mwðHÞ ¼mwð�HÞ � mon-mwð�HÞ ¼ htwð�HÞ ¼ ghtwðHÞ. &
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 293
Lemma 6.2. Let H be a hypergraph and � :¼ max�jhj
�� h 2 EðHÞ�
. Then,
cwðHÞ � � � mwðHÞ:
Proof. Replace each of the k marshals by � cops. &
The clique shows that the upper bound in Lemma 6.2 is sharp. On the other
hand, the n-cycle shows that the inequality cannot be replaced by an equality. Our
next aim is to bound monotone marshal-width by cop-width.
Let ðT ; �; �Þ be a winning strategy, r the root of T . ðT; �; �Þ is called weakly
monotone, if for all t 2 VðTÞnfrg the inclusion �ðtÞ � �ðpredðtÞÞ holds. Clearly,
every monotone winning strategy is weakly monotone. Conversely we have:
Lemma 6.3. Let H be a hypergraph. Every weakly monotone winning strategy
for MarðH; kÞ gives rise to a monotone winning strategy for �k marshals on H.
Proof. Let ðT; �; �Þ be a weakly monotone winning strategy for MarðH; kÞ,Call a vertex t 2 VðTÞ a stagnation vertex, if �ðtÞ ¼ �ðpredðtÞÞ. By contracting
ðt; predðtÞÞ for all stagnation vertices in VðTÞ, we obtain a monotone winning
strategy for �k marshals on H. &
Theorem 6.1. Let H be a hypergraph. Then mon-mwðHÞ � cwðHÞ.
Proof. Let cwðHÞ ¼ k and ðT ; �; �Þ be a monotone winning strategy for k
cops on H. Define a map ’ : VðHÞ ! EðHÞ with v 2 ’ðvÞ for all v 2 VðHÞ. Now
the k marshals can win as follows: in the beginning, they occupy the hyperedges
’ðvÞ with v 2 �ðrÞ; r the root of T . The robber chooses an escape space �0 with
respect to �0 :¼S
v2�ðrÞ ’ðvÞ. The escape space is a subset of �ðtÞ for exactly one
successor t of r. The marshals respond by occupying the hyperedges ’ðvÞ with
v 2 �ðtÞ, etc. Thus, we obtain a winning strategy ðT 0; �0; �0Þ together with a map
: VðT 0Þ ! VðTÞ, satisfying:
(1) For all t0 2 VðT 0Þnfr0g (where r0 is the root of T 0), ðpredðt0ÞÞ ¼predð ðtÞÞ and ðr0Þ ¼ r.
(2) For all t0 2 VðT 0Þ; �0ðt0Þ ¼ f’ðvÞ j v 2 �ð ðt0ÞÞg and �0ðt0Þ � �ð ðt0ÞÞ.
By Lemma 6.3, it is sufficient to show that this winning strategy is weakly
monotone. Suppose it is not. Then, there exists a t0 2 VðT 0Þnfr0g such that
�0ðt0Þ�=�0ðpredðt0ÞÞ. Hence, let v 2 �0ðt0Þn�0ðpredðt0ÞÞ. Note that predðt0Þ 6¼ r0,
because otherwise �0ðpredðt0ÞÞ ¼ VðHÞ. Therefore, pred2ðt0Þ is defined. While the
marshals fly from �0ðpred2ðt0ÞÞ to �0ðpredðt0ÞÞ, the escape spaces �0ðpredðt0ÞÞ and
�0ðt0Þ are connected, that is there exists a path W in VðHÞn�S
�0ðpred2
ðt0ÞÞ \S�0ðpredðt0ÞÞ
�from a vertex w 2 �0ðpredðt0ÞÞ to v. We may assume that w
is the last vertex in �0ðpredðt0ÞÞ on the path W . Let u be the vertex on W
immediately after w. Since u =2 �0ðpredðt0ÞÞ and w 2 �0ðpredðt0ÞÞ are neighbours,
it follows that u 2S�0ðpred2ðt0ÞÞ. This implies that u 2 ’ðxÞ for an x 2
294 JOURNAL OF GRAPH THEORY
�ðpred2ðtÞÞ, where t ¼ ðt0Þ. Since u lies on W ; u =2S�0ðpredðt0ÞÞ and hence
�ðxÞ =2 �0ðpredðt0ÞÞ. But then x =2 �ðpredðtÞÞ. Thus, we have found a non-
monotone move in the strategy for the cops.
In VðHÞn��ðpred2ðtÞÞ \ �ðpredðtÞÞ
�; x is connected to w 2 �0ðpredðt0ÞÞ �
�ðpredðtÞÞ via W and �ðxÞ. Since x =2 �ðpredðtÞÞ; x lies in a possible escape space
with respect to �ðpredðtÞÞ in the Robber and Cops Game. But x 2 �ðpred2ðtÞÞ, a
contradiction to the monotonicity of ðT ; �; �Þ. &
Finding a hypergraph G satisfying the equality mon-mwðGÞ ¼ cwðGÞ ¼ 3 is
left to the reader as an exercise.5 Putting things together we obtain.
Corollary 6.2. Let H be a hypergraph with maxfjhj j h 2 EðHÞg ¼ �. Then,
mwðHÞ� ghtwðHÞ ¼ htwð�HÞ� htwðHÞ ¼ mon-mwðHÞ� cwðHÞ ¼ mon-cw
ðHÞ ¼ twðHÞ þ 1 � � � mwðHÞ.None of the inequalities can be replaced by strict inqualities or by equalities.
7. CONCLUSIONS
Having presented a hypergraph with monotonicity cost n, a graph and a simpli-
cialised hypergraph with monotonicity cost 1, we leave as an open problem
whether there are graphs or simplicialised hypergraphs with monotonicity cost
greater than 1. The question whether there are structural characteristics disting-
uishing hypergraphs with monotonicity cost from others, is still open.
ACKNOWLEDGMENTS
The author thanks Prof. J. Flum, H. Scheuermann and A. Scivos for their valuable
suggestions on this paper.
REFERENCES
[1] I. Adler, Diploma thesis: Spiele als Hilfsmittel zu Strukturun tersuchungen
bei Graphen und Hypergraphen, Freiburg, 2002, http://www.math.uni-
freiburg.de/archiv/diplom/isolde_adler.html
[2] R. Diestel, Graph Theory, Springer, New York, 1997.
[3] J. Flum, M. Frick, and M. Grohe, Query evaluation via tree-decompositions,
J Assoc Comput Machin 49(6) (2002), 716–752.
[4] G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions and
tractable queries. J Comput Syst Sci 64(3) (2002), 579–627.
5Or see [1] for an example.
MARSHALS, MONOTONE MARSHALS, AND HYPERTREE-WIDTH 295
[5] G. Gottlob, N. Leone, and F. Scarcello. Robbers, Marshals, and Guards:
Game Theoretic and Logical Characterizations of Hypertree Width, Proceed-
ings of the Twentieth ACM Symposium on Principles of Database Systems,
ACM Press, New York, (2001), 21–32.
[6] P. D. Seymour and R. Thomas. Graphs searching and a min-max-theorem for
tree-width, J Combin Theory Series B 58 (1993), 22–33.
296 JOURNAL OF GRAPH THEORY