distance domination, guarding and covering of maximal outerplanar graphs
TRANSCRIPT
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Distance domination, guarding and covering of maximal
outerplanar graphs
Santiago Canales, Gregorio Hernández, Mafalda Martins, Inês Matos
Discrete Applied Mathematics 181 (2015) 41–49
報告者 :陳政謙
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Outline• Relationship between distance guarding, distance
domination and distance covering on triangulation graphs
• 2d-guarding and 2d-domination of maximal outerplanar graphs
• 2d-covering of maximal outerplanar graphs
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Triangulations
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The kd-visible• Given a triangulation T = (V, E), we say that a bounded face Ti
of T (i.e., a triangle) is kd-visible from a vertex p V, if there is a vertex x Ti such that distT(x, p) ≤ k − 1.
k = 1k = 2k = 3
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The kd-guarding set• A kd-guarding set for T is a subset F V such that every
triangle of T is kd-visible from an element of F (we designate the elements of F by kd-guards).
k = 2
The 2d-guarding number is 1.
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The kd-dominating set• A kd-dominating set for T is a subset D V such that each
vertex u V − D, distT(u, v) ≤ k for some v D.
The 2d-domination number is 1.
k = 2
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Lemma 2.1• If C is a kd-guarding set for a triangulation T, then C is a kd-
dominating set for T.k = 2
This is a 2d-guarding set. This is a 2d-dominating set.
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Lemma 2.1• If C is a kd-guarding set for a triangulation T, then C is a kd-
dominating set for T.k = 2
This is not a 2d-guarding set.This is a 2d-dominating set.
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The kd-vertex-cover• A kd-vertex-cover of T is a subset C V such that for each
edge e E there is a path of length at most k, which contains e and a vertex of C.
The 2d-covering number is 1.
k = 2
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Lemma 2.2• If C is a kd-vertex-cover of a triangulation T, then C is a kd-
guarding set and a kd-dominating set for T.k = 2
This is a 2d-guarding set.This is a 2d-vertex-cover. This is a 2d-dominating set.
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Lemma 2.2• If C is a kd-vertex-cover of a triangulation T, then C is a kd-
guarding set and a kd-dominating set for T.k = 2
This is a 2d-guarding set. This is not a 2d-vertex-cover.
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Lemma 2.2• If C is a kd-vertex-cover of a triangulation T, then C is a kd-
guarding set and a kd-dominating set for T.k = 2
This is not a 2d-vertex-cover.This is a 2d-dominating set.
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Theorem 2.3• Given a triangulation T, the minimum cardinality gkd(T) of any
kd-guarding set for T verifies γkd(T) ≤ gkd(T) ≤ βkd(T), where γkd(T) is the kd-domination number, gkd(T) is the kd-guarding number and βkd(T) is the kd-covering number.
• This theorem is proved by previous lemmas.
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Outerplanar graphs• A graph is outerplanar if it has a crossing-free embedding in
the plane such that all vertices are on the boundary of its outer face (the unbounded face).
Outer face Inner faceInner face
Inner face
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Outerplanar graphs• A graph is outerplanar if it has a crossing-free embedding in
the plane such that all vertices are on the boundary of its outer face (the unbounded face).
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Maximal outerplanar graphs• An outerplanar graph is said to be maximal outerplanar if
adding an edge removes its outerplanarity.
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Maximal outerplanar graphs – Property 1
• For n ≥ 3, every maximal outerplanar graph with n vertices has a cycle which all of the edges are on the boundary of the outer face.
……
…...
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Maximal outerplanar graphs – Property 2
• Every bounded face of maximal outerplanar graphs is a triangle.
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Maximal outerplanar graphs – Property 3
• Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges.
n = 8
2n - 3 = 13
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m = 6
f(m) 2d-guards• f(m) 2d-guards are always sufficient to guard any maximal
outerplanar graph with m vertices.
f(6) 2d-guards = 1
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m = 10
f(m) 2d-guards• f(m) 2d-guards are always sufficient to guard any maximal
outerplanar graph with m vertices.
f(10) 2d-guards = 2
…
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Lemma 3.1• If G is an arbitrary maximal outerplanar graph with two 2d-
guards placed at any two adjacent of its m vertices, then f(m − 2) additional 2d-guards are sufficient to guard G.
m = 8f(6) additional 2d-guards are sufficient to guard G.
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Lemma 3.2• If G is an arbitrary maximal outerplanar graph with one 2d-
guard placed at any one of its m vertices, then f(m − 1) additional 2d-guards are sufficient to guard G.
m = 7f(6) additional 2d-guards are sufficient to guard G.
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Lemma 3.3 ([13])• Let G be a maximal outerplanar graph with n ≥ 2k vertices.
There is an interior edge e in G that separates off a minimum number m of exterior edges, where k ≤ m ≤ 2k − 2. Therefore, the edge e partitions G into two pieces, one of which contains m exterior edges of G.n = 6
6 ≥ 63 ≤ m ≤ 4
k = 3
m = 3 m = 4
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Lemma 3.3 ([13])• Let G be a maximal outerplanar graph with n ≥ 2k vertices.
There is an interior edge e in G that separates off a minimum number m of exterior edges, where k ≤ m ≤ 2k − 2. Therefore, the edge e partitions G into two pieces, one of which contains m exterior edges of G.n = 8
8 ≥ 84 ≤ m ≤ 6
k = 4
m = 4 m = 5 m = 6
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Lemma 3.4• Every n-vertex maximal outerplanar graph, with 5 ≤ n ≤ 14,
can be 2d-guarded with guards.n = 6
= 1
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Lemma 3.4• Every n-vertex maximal outerplanar graph, with 5 ≤ n ≤ 14,
can be 2d-guarded with guards.n = 10
…
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The proof of Lemma 3.4• n = 5
Trivial. A 2d-guard can be placed at any vertex.
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The proof of Lemma 3.4• n = 6
……….…….…
Trivial. A 2d-guard can be placed at any vertex with degree greater than 2.
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The proof of Lemma 3.4• n = 7• Total edges = 2n – 3 = 11• Total degree = 11 x 2 = 22
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The proof of Lemma 3.4• n = 8
0
1 2
3
4
56
7
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The proof of Lemma 3.4• n = 9• By Lemma 3.3 with k = 4, there is an interior edge e that
separates off a minimum number m of exterior edges, where m = 4, 5 or 6.
m = 4
0
1
2 3
4
5
67
8
m = 5
0
1
2 3
4
5
67
8
m = 6
0
1
2 3
4
5
67
8
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The proof of Lemma 3.4• 10 ≤ n ≤ 14• The proof is like the previous case.
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Theorem 3.5• Every n-vertex maximal outerplanar graph, with n ≥ 5, can be
2d-guarded by guards.
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The proof of Theorem 3.5• The proof is done by induction on n.• By Lemma 3.4 the result is true for 5 ≤ n ≤ 14.• Assume that n ≥ 15, and the theorem holds for n′ < n.• Lemma 3.3 with k = 6 guarantees the existence of an interior
edge e that divides G into maximal outerplanar graphs G1 and G2, such that G1 has m exterior edges of G with 6 ≤ m ≤ 10.
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Thus G1 and G2 together can be 2d-guarded by guards.
n - m + 1 ≤ n – 5, by the induction hypothesis, G2 can be 2d-guarded by guards.
The proof of Theorem 3.5• 6 ≤ m ≤ 8
012
m - 2m - 1 m m + 1
m + 2
n - 2n - 1
e…… …
…
m + 1 ≤ 9, thus G1 can be 2d-guarded by one guard.
G1 G2
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The proof of Theorem 3.5• m = 9
The pentagon (5,6,7,8,9) can be 2d-guarded by placing one guard arbitrarily.To 2d-guard the hexagon (0,1,2,3,4,5), we cannot place a 2d-guard at any vertex. We will consider two separate cases.
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The proof of Theorem 3.5• m = 9
(a) The internal edge (0,4) is not present.If a guard is placed at vertex 5, then G1 is 2d-guarded.
Since G2 has n − 8 edges, the induction hypothesis can be applied and therefore G2 can be 2d-guarded by guards.
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The proof of Theorem 3.5• m = 9
(b) The internal edge (0,4) is present.If a 2d-guard is placed at vertex 0, then G1 is 2d-guarded unless the triangle (6,7,8) is present in the triangulation.In any case, two 2d-guards placed at vertices 0 and 9 guard G1.
By the induction hypothesis,guards suffice to 2d-guard G2.
By Lemma 3.1 the two 2d-guards placed at vertices 0 and 9 allow the remainder of G2 to be guarded by f (n−8−2) = f (n−10) additional 2d-guards.
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The proof of Theorem 3.5• m = 10
(a) The vertices 0 and 10 have degree 2 in hexagons (0,1,2,3,4,5) and (5,6,7,8,9,10), respectively.Then one 2d-guard placed at vertex 5 guards G1.By the induction hypothesis, G2 can be 2d-guarded by guards.
Thus G can be 2d-guarded by guards.
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The proof of Theorem 3.5• m = 10
(b) The vertex 0 has degree greater than 2 in hexagon (0,1,2,3,4,5).We place a guard at vertex 0 and another guard at any vertex of the hexagon (5,6,7,8,9,10) whose degree is greater than 2.These two guards 2d-guard G1.By Lemma 3.2 the guard placed at vertex 0 permits the remainder of G2 to be 2d-guarded by f (n−9−1) = f (n−10) additional guards.By the induction hypothesis, guards suffice to 2d-guard G2.
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Theorem 3.6• Every n-vertex maximal outerplanar graph with n ≥ 5 can be
2d-guarded (and 2d-dominated) by guards. This bound is tight.
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According to Theorem 2.3, γ2d(G) ≤ g2d(G), so ≤ g2d(G),where γ2d(G) is the 2d-domination number, g2d(G) is the 2d-guarding number.
The proof of Theorem 3.6
The 2d-domination number is .
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Lemma 4.1 ([17])• The vertices of a maximal outerplanar graph G can be 4-
colored such that every cycle of length four in G has all four colors.
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Theorem 4.2• Every n-vertex maximal outerplanar graph, with n ≥ 4, can be
2d-covered by vertices and this bound is tight.n = 5
= 1
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The proof of Theorem 4.2• First, we show that every maximal outerplanar graph has a 2d-
covering with vertices.• By Lemma 4.1, we can color the vertices of G such that every
cycle of length four is 4-colored.
…....…
……....…..
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The proof of Theorem 4.2• Now, we will prove that this upper bound is tight.
β2d(G) ≥ , where β2d(G) is the 2d-covering number.
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References