In this paper we de ne a distance guarding concept on plane graphs and associate this concept with distance domination and distance vertex cover concepts on triangulation graphs. Furthermore, for any n-vertex maximal outerplanar graph, we provide tight upper bounds for g2d(n) (2d-guarding number), γ2d(n) (2d-domination number) and β2d(n) (2d-vertex cover number).European Science FoundationMinisterio de Ciencia e InnovaciónFundação para a Ciência e a TecnologiaFondo Europeo de Desarrollo Regiona

Canales Cano, Santiago

Díaz Báñez, José Miguel (Coordinador)

Garijo Royo, Delia (Coordinador)

Hernández Peñalver, Gregorio

Matos. Inês

Márquez Pérez, Alberto (Coordinador)

Oliveira Martins, Ana Mafalda de

Urrutia Galicia, Jorge (Coordinador)

English

idUS. Depósito de Investigación Universidad de Sevilla

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Distance domination, guarding and vertex cover for maximalouterplanar graphsSantiago Canales1, Gregorio Hernández∗2, Mafalda Martins‡†3, and Inês Matos‡31Universidad Ponticia Comillas de Madrid, Spain2Universidad Politécnica de Madrid, Spain3Universidade Aveiro & CIDMA, PortugalAbstractIn this paper we dene a distance guarding concept onplane graphs and associate this concept with distancedomination and distance vertex cover concepts on tri-angulation graphs. Furthermore, for any n-vertexmaximal outerplanar graph, we provide tight up-per bounds for g2d(n) (2d-guarding number), γ2d(n)(2d-domination number) and β2d(n) (2d-vertex covernumber).IntroductionDomination, covering and guarding are widely studiedsubjects in graph theory. Given a graph G = (V,E)a dominating set is a subset D of V such that ev-ery vertex not in D is adjacent to a vertex in D. Asubset C of V is a vertex cover if each edge of thegraph is incident to at least one vertex of the set.Thus, a dominating set guards the vertices of a graphwhile a vertex cover guards its edges. In plane graphsthese concepts dier from the guarding set conceptthat guards the faces of the graph. Let G = (V,E) bea plane graph, a guarding set is a subset S of V suchthat every face has a vertex in S. There are many pa-pers and books about domination and its many vari-ants in graphs, e.g. [4, 8, 9, 10]. Domination was ex-tended to distance domination by Meir and Moon [11]:given a graph G, a subset D of V is said to be a dis-tance k-dominating set if for each vertex u ∈ V −D,distG(u, v) ≤ k for some v ∈ D. The minimum car-dinality of a distance k-dominating set is said to bethe distance k-domination number of G and is de-noted by γkd(G). In the case of distance domination,∗Research supported by ESF EUROCORES programme Eu-roGIGA - ComPoSe IP04 - MICINN Project EUI-EURC-2011-4306.†Email: mafalda.martins@ua.pt. Research also supportedby FCT grant SFRH/BPD/66431/2009.‡Research supported by FEDER funds throughCOMPETEOperational Programme Factors of Com-petitiveness, CIDMA and FCT within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.there are also some known results concerning boundsfor γkd(G), e.g., [13, 14]. However, if graphs are re-stricted to triangulations, as far as we know, thereare no known bounds for γkd(G). The distance domi-nation was generalized to broadcast domination whenthe power of each vertex may vary [6]. Given a graphG = (V,E), a broadcast is a function f : V → N0. Abroadcast is dominating if for every vertex v, thereis a vertex u such that f(u) > 0 and d(u, v) ≤ f(u).A dominating broadcast f is optimal if∑v∈V f(v)is minimum over all choices of broadcast dominatingfunctions for G. The broadcast domination problemconsists in building this optimal function. Note that,if f(V ) = {0, k}, then the broadcast domination prob-lem is the distance k-dominating problem. If a broad-cast f provides coverage to the edges of G instead ofcovering its vertices, we have a generalization of thevertex cover concept [2]. A broadcast f is covering iffor every edge (x, y) ∈ E, there is a path P in G thatincludes the edge (x, y) and one end of P must be avertex u, where f(u) is at least the length of P . A cov-ering broadcast f is optimal if∑v∈V f(v) is minimumover all choices of broadcast covering functions for G.Note that, if f(V ) = {0, 1}, then the broadcast coverproblem coincides with the problem of nding a min-imum vertex cover. Regarding the broadcast coverproblem when all vertices have the same power (i.e.,when f(V ) = {0, k}, for a xed k ̸= 1), as far as weknow, there are no published results besides [5] wherethe authors propose a centralized and distributed ap-proximation algorithm to solve it. Concerning theguarding concept there are also known bounds on theguarding number g(G). For example, g(G) ≤ ⌊n2 ⌋, forany n-vertex plane graph [3], and if G is a maximalouterplanar graph this bound is ⌊n3 ⌋ [7]. Contrary tothe notions of domination and vertex cover on planegraphs that were extended to include its distance ver-sions, the guarding concept was not generalized to itsdistance version.In this paper we generalize the guarding concepton plane graphs to its distance guarding version. Wealso formalize the broadcast cover problem when allvertices have the same power, which we call distance51Distance domination, guarding and vertex cover for maximal outerplanar graphsk-vertex cover. Since there are no combinatorial re-sults for the concepts of domination, vertex coverand guarding on its distance versions on triangulationgraphs (triangulations, for short), we also address thisproblem. In the next section we describe some of theterminology that will be used throughout this paper.1 PreliminariesGiven a triangulation of a point set T = (V,E), wesay 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 ∈ Tisuch that distT (x, p) ≤ k − 1. The kd-visibility regionof a vertex p ∈ V comprises the triangles of T that arekd-visible from p (see Fig. 1).p p( )a (b)Figure 1: The kd-visible region of p for: (a) k = 1;(b) k = 2.A kd-guarding set for T is a subset F ⊆ V such thatevery triangle of T is kd-visible from an element of F .We designate the elements of F by kd-guards. The kd-guarding number gkd(T ) is the number of vertices ina smallest kd-guarding set for T . Note that, to avoidconfusion with multiple guarding [1] - where the typ-ical notation is k-guarding - we will use kd-guarding,with an extra d. Given a set S and a positiveinteger n, we dene gkd(S) = max{gkd(T ) : T =(V,E) is triangulation with V = S} and gkd(n) =max{gkd(S) : |S| = n}. A kd-vertex cover for T , ordistance k-vertex cover for T , is a subset C ⊆ V suchthat for each edge e ∈ E there is a path of length atmost k, which contains e and a vertex of C. Thekd-vertex cover number βkd(T ) is the number of ver-tices in a smallest kd-vertex cover set for T . Givena set S and a positive integer n, we dene βkd(S) =max{βkd(S) : T = (V,E) is triangulation with V =S} and βkd(n) = max{βkd(S) : |S| = n}. Finally,as already dened by other authors, a kd-dominatingset for T , or distance k-dominating set for T , isa subset D ⊆ V such that each vertex u ∈ V −D,distT (u, v) ≤ k for some v ∈ D. The kd-dominationnumber γkd(T ) is the number of vertices in a smallestkd-dominating set for T . Given a set S and a positiveinteger n, we dene γkd(S) = max{γkd(T ) : T =(V,E) is triangulation with V = S} and γkd(n) =max{γkd(S) : |S| = n}. Our main goal is to obtainbounds on γkd(T ), gkd(T ) and βkd(T ). We start byestablishing a tight upper bound for g2d(n) for a spe-cial class of triangulations, namely the maximal out-erplanar graphs. A graph is a maximal outerplanargraph if it is a triangulation of a simple polygon with-out holes [12]. Edges on the exterior face are calledexterior edges, and interior edges otherwise.In the next section we show that there is a re-lationship between 2d-guarding, 2d-dominating and2d-vertex cover sets on triangulations. In sections 3and 4 we provide upper bounds for g2d(n), γ2d(n) andβ2d(n) on maximal outerplanar graphs and show thatthese bounds are tight.2 Relationship between distancevertex cover, guarding anddomination on triangulationsWe start by showing that the three concepts are dif-ferent. Fig. 2 depicts 2d-dominating and 2d-guardingsets for a given triangulation. Note that in Fig. 2(a)the set {u, v} is 2d-dominating since the remainingvertices are at distance 1 or 2. However, it is nota 2d-guarding set because the shaded triangle is notguarded, as its vertices are at distance 2 from {u, v}.In 2(b) {w, z} is a 2d-guarding set, however it is not a2d-vertex cover since any path between the bold edgeand w or z has length at least 3. Therefore, the boldedge is not covered.uvwz( )a (b)Figure 2: (a) 2d-dominating set for a triangulation T ;(b) 2d-guarding set for T .Now we are going to establish a relation betweeng2d(T ), γ2d(T ) and β2d(T ). The following results canbe easily generalized to gkd(T ), γkd(T ) and βkd(T ).Lemma 1 If C is a 2d-vertex cover of a triangulationT , then C is a 2d-guarding set and a 2d-dominatingset of T .Lemma 2 If F is a 2d-guarding set of a triangulationT , then F is a 2d-dominating set of T .The previous lemmas prove the following result.Theorem 3 Given a triangulation T the minimumcardinality g2d(T ) of any 2d-guarding set veriesγ2d(T ) ≤ g2d(T ) ≤ β2d(T ).52XV Spanish Meeting on Computational Geometry, June 26-28, 2013(a) (b) (c)Figure 3: (a) A 2d-dominating set for a triangulation T (black vertices); (b) a 2d-guarding set for T (grayvertices); (c) each of the bold edges needs a dierent vertex to be 2d-covered.Note that the inequalities above can be strict, aswe can see in the triangulation T presented in Fig. 3,where γ2d(T ) = 2, g2d(T ) = 3, and β2d(T ) ≥ 4.3 2d-domination and 2d-guarding of maximal out-erplanar graphsIn this section we establish upper bounds for g2d(n)and γ2d(n) on maximal outerplanar graphs. In orderto do this, and following the ideas of O'Rourke [12],we rst need to introduce some lemmas.Lemma 4 Suppose that f(m) guards are always suf-cient to 2d-guard any outerplanar maximal graph Gwith m vertices. Then if G has two guards placedat any two adjacent vertices or one guard placed atany one of its vertices, then f(m− 2) or f(m− 1)additional guards are sucient to 2d-guard G, respec-tively.Lemma 5 Let G be an outerplanar maximal graphwith n ≥ 2k vertices. There is an interior edge eof G that partitions G into two pieces, one of whichcontains m = k, k + 1, . . . , 2k − 1 or 2k − 2 exterioredges of G.Theorem 6 Every n-vertex maximal outerplanargraph, with n ≥ 5, can be 2d-guarded by ⌊n5 ⌋ 2d-guards. That is, g2d(n) ≤ ⌊n5 ⌋ for all n ≥ 5.Proof. For 5 ≤ n ≤ 11, the truth of the theoremcan be easily established. It should be noted thatfor n = 5 the 2d-guard can be placed randomly andfor n = 6 it can be placed at any vertex of degreegreater than 2. Assume that n ≥ 12 and that thetheorem holds for all n′ < n. Lemma 5 guaranteesthe existence of an interior edge e that divides G intotwo maximal outerplanar graphsG1 andG2, such thatG1 has m exterior edges of G with 6 ≤ m ≤ 10. Thevertices of G are labeled with 0, 1, . . . , n− 1 such thate is (0,m). Each value of m is considered separately.We present the cases m = 6 and m = 9.Case m = 6. G1 has m+ 1 = 7 exterior edges, thusG1 can be 2d-guarded with one guard. G2 has n − 5exterior edges including e, and by induction hypothe-sis, it can be 2d-guarded with ⌊n−55 ⌋ = ⌊n5 ⌋−1 guards.Thus G1 and G2 together can be 2d-guarded by ⌊n5 ⌋guards.Case m = 9. The presence of any of the internal edges(0,8), (0,7), (0,6), (9,1), (9,2) and (9,3) would violatethe minimality of m. Thus, the triangle T in G1 thatis bounded by e is either (0,5,9) or (0,9,4). Since theseare equivalent cases, suppose that T is (0,5,9) (see Fig.4(a)).0 97654321 8eG2TG10 976548eG2321TG1( )a ( )bFigure 4: (a) The triangle T is (0,5,9); (b) the internaledge (0,4) and the triangle (6,7,8) are present.The pentagon (5,6,7,8,9) can be 2d-guarded by plac-ing one guard at a chosen vertex. However, to 2d-guard the hexagon (0,1,2,3,4,5) we cannot place aguard randomly.We will consider two separate cases:(i) The internal edge (0,4) is not present: if a guardis placed at vertex 5, then the hexagon (0,1,2,3,4,5)is 2d-guarded, thus G1 is 2d-guarded. Since G2 hasn− 8 edges it can be 2d-guarded by ⌊n−85 ⌋ ≤ ⌊n5 ⌋ − 1guards applying the induction hypothesis. This yieldsa 2d-guarding of G by ⌊n5 ⌋ guards; (ii) The internaledge (0,4) is present: if a guard is placed at vertex0, then G1 is 2d-guarded unless the triangle (6,7,8)is present in the triangulation (see Fig. 4(b)). Inany case, two 2d-guards placed at vertices 0 and 9guard G1. G2 has n − 8 exterior edges, includinge. By lemma 4 the two guards placed at vertices 0and 9 allow the remainder of G2 to be guarded byf(n− 8− 2) = f(n− 10) additional 2d-guards. Recallthat f(n′) is the number of 2d-guards that are alwayssucient to guard a maximal outerplanar graph with53Distance domination, guarding and vertex cover for maximal outerplanar graphsn′ vertices. By the induction hypothesis f(n′) = ⌊n′5 ⌋.Thus, ⌊n−105 ⌋ = ⌊n5 ⌋ − 2 guards suces to 2d-guardG2. Together with the guards placed at vertices 0and 9 that 2d-guard G1, all G is guarded by ⌊n5 ⌋ 2d-guards.To prove that this upper bound is tight we needto construct a maximal outerplanar graph G of or-der n such that g2d(G) = ⌊n5 ⌋. Fig. 5 shows a maxi-mal outerplanar graph G for which γ2d(G) =n5 , sincethe black vertices can only be 2d-dominated by dif-ferent vertices. Thus, γ2d(n) ≥ n5 . Note that thisexample can be generalized to kd-domination to ob-tain γkd(n) ≥ n(2k+1) .Figure 5: A maximal outerplanar graph G for whichγ2d(G) =n5 .According to theorem 3, γ2d(G) ≤ g2d(G), so⌊n5 ⌋ ≤ g2d(G). In conclusion, ⌊n5 ⌋ 2d-guards are oc-casionally necessary and always sucient to guard an-vertex maximal outerplanar graph G. On the otherhand, we can also establish that γ2d = ⌊n5 ⌋, since⌊n5 ⌋ ≤ γ2d(n) and γ2d(n) ≤ g2d(n), for all n. Thus,it follows:Theorem 7 Every n-vertex maximal outerplanargraph with n ≥ 5 can be 2d-guarded (and 2d-dominated) by ⌊n5 ⌋ guards. This bound is tight.4 2d-covering of maximal outer-planar graphsIn this section we determine an upper bound for 2d-vertex cover on maximal outerplanar graphs which isalso tight.Lemma 8 Suppose that f(m) vertices are always suf-cient to 2d-cover any outerplanar maximal graph Gwith m vertices. If we randomly choose a vertex of Gto be a 2d-covering vertex, then f(m − 1) additionalvertices are sucient 2d-cover G.Theorem 9 Every n-vertex maximal outerplanargraph, with n ≥ 4, can be 2d-covered with ⌊n4 ⌋ ver-tices. That is, β2d(n) ≤ ⌊n4 ⌋ for all n ≥ 4.Now, we will prove that this upper bound is tight.The bold edges of the maximal outerplanar graph il-lustrated in Fig. 6 can only be 2d-covered from dier-ent vertices, and therefore β2d(n) ≥ n4 . To conclude:Theorem 10 Every n-vertex maximal outerplanargraph with n ≥ 5 can be 2d-covered by ⌊n4 ⌋ vertices.This bound is tight.Figure 6: A maximal outerplanar graph G for whichβ2d(G) =n4 .References[1] P. Belleville, P. Bose, J. Czyzowicz, J. Urrutia, J.Zaks, K-vertex guarding simple polygons, Comput.Geom. Theory Appl. , 42(4) (2009), 352361.[2] J.R.S. Blair and S.B. Horton, Broadcast covers ingraphs, Congr. Num. 173 (2005), 109115.[3] P. Bose, T. Shermer, G. Toussaint, and B. 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Distance domination, guarding and vertex cover for maximal outerplanar graphs

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Distance domination, guarding and vertex cover for maximal outerplanar graphs

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