graphs represented by words
DESCRIPTION
Graphs represented by words. Joint work with. Magnus M. Halldorsson. Sergey Kitaev Reykjavik University. Reykjavik University. Artem Pyatkin. Sobolev Institute of Mathematics. Basic definitions. A finite word over { x , y } is alternating if it does not contain xx and yy. - PowerPoint PPT PresentationTRANSCRIPT
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Graphs represented by words Sergey Kitaev
Reykjavik University
Sobolev Institute of Mathematics
Joint work with
Artem Pyatkin
Magnus M. Halldorsson
Reykjavik University
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Basic definitions
Sergey Kitaev Graphs represented by words
A finite word over {x,y} is alternating if it does not contain xx and yy.
Alternating words: yx, xy, xyxyxyxy, yxy, etc.
Non-alternating words: yyx, xyy, yxxyxyxx, etc.
Letters x and y alternate in a word w if they induce an alternating subword.
x and y alternate in w = xyzazxayxzyax
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Basic definitions
Sergey Kitaev Graphs represented by words
A finite word over {x,y} is alternating if it does not contain xx and yy.
Alternating words: yx, xy, xyxyxyxy, yxy, etc.
Non-alternating words: yyx, xyy, yxxyxyxx, etc.
Letters x and y alternate in a word w if they induce an alternating subword.
x and y alternate in w = xyzazxayxzyax
x and y do not alternate in w = xyzazyaxyxzyax
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Basic definitions
Sergey Kitaev Graphs represented by words
A word w is k-uniform if each of its letters appears in w exactly k times.
A 1-uniform word is also called a permutation.
A graph G=(V,E) is represented by a word w if 1. Var(w)=V, and2. (x,y) E iff x and y alternate in w.
word-representant
A graph is (k-)representable if it can be represented by a (k-uniform) word.
A graph G is 1-representable iff G is a complete graph.
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Example of a representable graph
Sergey Kitaev Graphs represented by words
cycle graph
x
y
v
z a
xyzxazvay represents the graph
xyzxazvayv 2-represents the graph
Switching the indicated x and a would create an extra edge
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Sergey Kitaev Graphs represented by words
Cliques and Independent Sets
Kn
Clique
Kn
Independent set
W=ABC...Z ABC...Z W=ABC...YZ ZY...CBA
V={A,B,C,...Z}
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Sergey Kitaev Graphs represented by words
Original motivation to study such representable graphs: The Perkins semigroup
S. Kitaev, S. Seif: Word problem of the Perkins semigroup via directed acyclic graphs, Order (2009).
Related work: Split-pair arrangement (application:
scheduling robots on a path, periodically)R. Graham, N. Zhang: Enumerating split-pair arrangements, J. Combin. Theory A, Feb. 2008.
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Sergey Kitaev Graphs represented by words
Papers on representable graphs:
S. Kitaev, A. Pyatkin: On representable graphs, Automata, Languages and Combinatorics (2008).
M. Halldorsson, S. Kitaev, A. Pyatkin: On representable graphs, semi-transitive orientations, and the representation numbers, preprint.
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Operations Preserving Representability
• Replacing a node v by a module H– H can be any clique or any comparability graph– Neighbors of v become neighbors of all nodes in H
• Gluing two representable graphs at 1 node
• Joining two representable graphs by an edge
Sergey Kitaev Graphs represented by words
G H+ = H G
G H& = G H
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Operations Not Preserving Representability
• Taking the line graph
• Taking the complement
• Attaching two graphs at more than 1 node
Sergey Kitaev Graphs represented by words
G H+ = H G
Open question: Does it preserve non-representability?
The graph in red is not 2- or 3-representable. It is not known if it is representable or not.
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Properties of representable graphs
Sergey Kitaev Graphs represented by words
If G is k-representable and m>k then G is m-representable.
For representable graphs, we may restrict ourselves to connected graphs.
G U H (G and H are two connected components) is representable iffG and H are representable. (Take concatenation of the corresponding words representants having at least two copies of each letter.)
If G is representable then G is k-representable for some k.
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2-representable graphs
• 1-representable graphs cliques• 2-representable graphs ??
• A B C D E F G H C D H G F A B D
Sergey Kitaev Graphs represented by words
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2-representable graphs
• View as overlapping intervals: u & v adjacent if they overlap
Example:
A B C D E F G H C D H G F A B E
Sergey Kitaev Graphs represented by words
E
A
F
u vuv E
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2-representable graphs
• View as overlapping intervals:
Equivalent to Interval overlap graphs
A B C D E F G H C D H G F A B E
Sergey Kitaev Graphs represented by words
E
A
F
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2-representable graphs
Sergey Kitaev Graphs represented by words
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2-representable graphs
Sergey Kitaev Graphs represented by words
Circle graphs
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Comparability graphs
• We can orient the edges to form a transitive digraph
• They correspond to partial orders.
Sergey Kitaev Graphs represented by words
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Comparability graphs
• We can orient the edges to form a transitive digraph
• They correspond to partial orders.
Sergey Kitaev Graphs represented by words
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Comparability graphs
• We can orient the edges to form a transitive digraph
• They correspond to partial orders.
Sergey Kitaev Graphs represented by words
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Representing comparability graphs
1. Form a topological ordering, where a given letter, say c, is as early as possible: abcdefg
Sergey Kitaev Graphs represented by words
e
b
c
a
g
fd
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Representing comparability graphs
1. Form a topological ordering, where a given letter, say c, is as early as possible: abcdefg
2. Then add another where it is as late as possible abfgdce
3. Repeat from 1. until done
Sergey Kitaev Graphs represented by words
e
b
c
a
g
fd
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Representing comparability graphs
1. The resulting substring abcdefg abfgdcecovers all non-edges incident on c.
Sergey Kitaev Graphs represented by words
e
b
c
a
g
fd
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Representing comparability graphs
1. The resulting substring abcdefg abfgdcecovers all non-edges incident on c.
2. For this graph it would suffice to repeat this for f: abfgcde abcdefgplus one round for d: dabcdfg
3. Final string:
Sergey Kitaev Graphs represented by words
e
b
c
a
g
fd
abcdefg abfgdce abfgcde abcdefg dabcdfg
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Properties of representable graphs
Sergey Kitaev Graphs represented by words
A graph is permutationally representable if it can be represented by a word of the form P1P2...Pk where Pis are permutations of the same set.
1
2
3
4
is permutationally representable (13243142)
Lemma (Kitaev and Seif). A graph is permutationally representable iff it is transitively orientable, i.e. if it is a comparability graph.
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Shortcut – a type of digraph
• Acyclic, non-transitive• Contains directed cycle
a, b, c, d, except last edge is reversed
• Non-transitive Not representable
Sergey Kitaev Graphs represented by words
d
b
c
a Missing!
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Main result
• A graph G is representable iff G is orientable to a shortcut-free digraph
• () Straightforward. • () We give an algorithm that takes any shortcut-
free digraph and produces a word that represents the graph
Sergey Kitaev Graphs represented by words
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Sketch of our algorithm
• Chain together copies of the digraph (= D’)– If ab D, then biai+1 D’
Sergey Kitaev Graphs represented by words
b c
d
a
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Sketch of our algorithm
• Chain together copies of the digraph (= D’)– If ab D, then biai+1 D’
Sergey Kitaev Graphs represented by words
b c
d
a
b c
d
a
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Sketch of our algorithm
• Chain together copies of the digraph (= D’)– If ab D, then biai+1 D’
• Form a topsort of D’ of pairs of copies.– In 1st copy, some letter d occurs as
late as possible– In 2nd copy d occurs as early as
possible
Sergey Kitaev Graphs represented by words
b c
d
a
b c
d
a
a b c a d c b dExample:
We allow the topsort to traverse the 2nd copy before finishing the 1st . The added edges ensure that adjacent nodes still alternate.
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Size of the representation
• The algorithm creates a word where each of the n letters appears at most n times.
Each representable graph is n-representable• There are graphs that require n/2 occurrences
– E.g. based on the cocktail party graph
• Deciding whether a given graph is k-representable, for k between 3 and [n/2], is NP-complete
Sergey Kitaev Graphs represented by words
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Corollary: 3-colorable graphs
• 3-colorable graphs are representable
• Red->Green->Blue orientation is shortcut-free!Sergey Kitaev Graphs represented by words
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Non-representable graphs
Sergey Kitaev Graphs represented by words
Lemma. Let x be a vertex of degree n-1 in G having n nodes. Let H=G \ {x}. Then G is representable iff H is permutationally representable.
The lemmas give us a method to construct non-representable graphs.
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Construction of non-representable graphs
Sergey Kitaev Graphs represented by words
1. Take a graph that is not a comparability graph (C5 is the smallest example);
2. Add a vertex adjacent to every node of the graph;3. Add other vertices and edges incident to them (optional).
W5 – the smallest non-representable graph
All odd wheels W2t+1 for t ≥ 2are non-representable graphs.
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Small non-representable graphs
Sergey Kitaev Graphs represented by words
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Relationships of graph classes
Sergey Kitaev Graphs represented by words
Representable
Circle 2-repres.
3-colorable Comparability
Bipartite2-inductive
Partial 2-trees
Outerplanar
2-outerplanar
3-trees
Trees
Chordal
2-trees
Perfect
Planar
4-colorable & K4-free
Split
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Sergey Kitaev Graphs represented by words
A property of representable graphs
G representable For each x V, G[N(x)] is permutationally representable,
Natural question: Is the converse statement true?
G[N(x)] = graph induced by the neighborhood of x
Main means of showing non-representability
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A non-representable graph whose induced neighborhood graphs are all comparability
Sergey Kitaev Graphs represented by words
co-T2 T2
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3-representable graphs
Sergey Kitaev Graphs represented by words
examples of prisms
Theorem (Kitaev, Pyatkin). Every prism is 3-representable.
Theorem (Kitaev, Pyatkin). For every graph G there exists a 3-representable graph H that contains G as a minor.
In particular, a 3-subdivision of every graph G is 3-representable.
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Sergey Kitaev Graphs represented by words
One more result
We can construct graphs with represntation number k=[n/2]
Coctail party graph:
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Sergey Kitaev Graphs represented by words
One more result
We can construct graphs with represntation number k=[n/2]
Coctail party graph:
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Complexity
• Recognizing representable graphs is in NP– Certificate is an orientation– Is it NP-hard?
• Most optimization problems are hard– Ind. Set, Dom. Set, Coloring, Clique Partition...
• Max Clique is polynomially solvable on repr.gr.– A clique is contained within some neighborhood– Neighborhoods induce comparability graphs
Sergey Kitaev Graphs represented by words
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• Is it NP-hard to decide whether a given graph is representable?
• What is the maximum representation number of a graph (between n/2 and n)?
• Can we characterize the forbidden subgraphs of representative graphs?
• Graphs of maximum degree 4? • How many (k-)representable graphs are there?
Sergey Kitaev Graphs represented by words
Open problems
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Sergey Kitaev Graphs represented by words
Resolved question
Is the Petersen’s graph representable?
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Sergey Kitaev Graphs represented by words
Resolved question
Is the Petersen’s graph representable?
It is 3-representable:
1
2
34
9 8
7
6
5 10
1,3,8,7,2,9,6,10,7,4,9,3,5,4,1,2,8,3,10,7,6,8,5,10,1,9,4,5,6,2
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Sergey Kitaev Graphs represented by words
Resolved questions
Are there any non-representable graphs with N(v) inducing a comparability graphs for every vertex v? In particular,– Are there non-representable graphs of maximum
degree 3?– Are there 3-chromatic non-representable graphs?– Are there any triangle-free non-representable
graphs?
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Sergey Kitaev Graphs represented by words
Open/Resolved problems
• Is it NP-hard to determine whether a given graph is NP-representable.
• Is it true that every representable graph is k-representable for some k?
• How many (k-)representable graphs on n vertices are there?
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Sergey Kitaev Graphs represented by words
Thank you for your attention!
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