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

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

Triangulations

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

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.

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

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.

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.

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

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.

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.

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.

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.

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

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).

Maximal outerplanar graphs• An outerplanar graph is said to be maximal outerplanar if

adding an edge removes its outerplanarity.

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.

……

…...

Maximal outerplanar graphs – Property 2

• Every bounded face of maximal outerplanar graphs is a triangle.

Maximal outerplanar graphs – Property 3

• Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges.

n = 8

2n - 3 = 13

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

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

…

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.

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.

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

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

Lemma 3.4• Every n-vertex maximal outerplanar graph, with 5 ≤ n ≤ 14,

can be 2d-guarded with guards.n = 6

= 1

Lemma 3.4• Every n-vertex maximal outerplanar graph, with 5 ≤ n ≤ 14,

can be 2d-guarded with guards.n = 10

…

The proof of Lemma 3.4• n = 5

Trivial. A 2d-guard can be placed at any vertex.

The proof of Lemma 3.4• n = 6

……….…….…

Trivial. A 2d-guard can be placed at any vertex with degree greater than 2.

The proof of Lemma 3.4• n = 7• Total edges = 2n – 3 = 11• Total degree = 11 x 2 = 22

The proof of Lemma 3.4• n = 8

0

1 2

3

4

56

7

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

The proof of Lemma 3.4• 10 ≤ n ≤ 14• The proof is like the previous case.

Theorem 3.5• Every n-vertex maximal outerplanar graph, with n ≥ 5, can be

2d-guarded by guards.

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.

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

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.

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.

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.

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.

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.

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.

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 .

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.

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

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.

…....…

……....…..

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.

References