In this paper we deal with two types of graph convexities, which are the most natural path convexities in a graph and which are defined by a system P of paths in a connected graph G: the geodesic convexity (also called metric
convexity) which arises when we consider shortest paths, and the monophonic convexity (also called minimal path convexity) when we consider chordless paths. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, showing that the contour must be monophonic. Finally, we consider the so-called edge Steiner sets. We prove that every edge Steiner set is edge monophonic.Ministerio de Ciencia y TecnologíaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny

Hernando Martin, Maria del Carmen

Mora Gine, Mercè

Pelayo Melero, Ignacio Manuel

Seara Ojea, Carlos

idUS. Depósito de Investigación Universidad de Sevilla

On geodesic and monophonic convexity 1Carmen Hernando a Merce` Mora b Ignacio M. Pelayo c Carlos Seara baDpt. Matema`tica Aplicada I, UPC, Barcelona, Spain, carmen.hernando@upc.esbDpt. Matema`tica Aplicada II, UPC, Barcelona, Spain, {merce.mora,carlos.seara}@upc.escDpt. Matema`tica Aplicada III, UPC, Barcelona, Spain, ignacio.m.pelayo@upc.esAbstractIn this paper we deal with two types of graph convexities, which are the most natural path convexities in a graphand which are defined by a system P of paths in a connected graph G: the geodesic convexity (also called metricconvexity) which arises when we consider shortest paths, and the monophonic convexity (also called minimal pathconvexity) when we consider chordless paths. First, we present a realization theorem proving, that there is nogeneral relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, showingthat the contour must be monophonic. Finally, we consider the so-called edge Steiner sets. We prove that everyedge Steiner set is edge monophonic.1. IntroductionA convexity on a finite set V is a family C ofsubsets of V , to be regarded as convex sets, which isclosed under intersection and which contains bothV and the empty set. The pair (V, C) is called aconvexity space. A finite graph-convexity space isa pair (G, C), formed by a finite connected graphG = (V,E) and a convexity C on V such that (V, C)is a convexity space satisfying that every memberof C induces a connected subgraph of G [4,5]. Thus,classical convexity can be extended to graphs in anatural way. We know that a set X of Rn is convexif every segment joining two points of X is entirelycontained in it. Similarly, a vertex set W of a finiteconnected graph G is said to be a convex set of Gif it contains all the vertices lying in a certain kindof path connecting vertices of W .In this paper we deal with two types of graphconvexities, which are the most natural path con-vexities in a graph and which are defined by asystem P of paths in G: the geodesic convexity(also called metric convexity) [5,6,7,11] whicharises when we consider shortest paths, and themonophonic convexity (also called minimal pathconvexity) [4,5] when we consider chordless paths.1 Partially supported by projects MCYT-FEDER TIC-2001-2171, MCYT-FEDER BFM 2002-0557, Gen Cat2001SGR00224, MCYT BFM2000-1052-C02-01.In what follows, G = (V,E) denotes a finite con-nected graph with no loops or multiple edges. Thedistance d(u, v) between two vertices u and v is thelength of a shortest u− v path in G. A chord of apath u0u1 . . . uh is an edge uiuj , with j ≥ i + 2. Au− v path ρ is called monophonic if it is a chord-less path, and geodesic if it is a shortest u−v path,that is, if |E(ρ)| = d(u, v).The geodesic closed interval I[u, v] is the set ofvertices of all u−v geodesics. Similarly, the mono-phonic closed interval J [u, v] is the set of verticesof all monophonic u − v paths. For W ⊆ V , thegeodesic closure I[W ] of W is defined as the unionof all geodesic closed intervals I[u, v] over all pairsu, v ∈ W . The monophonic closure J [W ] is the setformed by the union of all monophonic closed in-tervals J [u, v].A vertex set W ⊆ V is called geodesically convex(or simply g-convex ) if I[W ] = W , while it is saidto be geodetic if I[W ] = V . Likewise, W is calledmonophonically convex (or simply m-convex ) ifJ [W ] = W , and is called monophonic if J [W ] =V . The smallest g-convex set containing W is de-noted [W ]g and is called the g-convex hull of W .Similarly, the m-convex hull [W ]m of W is definedas the minimum m-convex set containing W . Ob-serve that J [W ] ⊆ [W ]m, I[W ] ⊆ [W ]g and [W ]g ⊆[W ]m. A g-hull (m-hull) set of G is a vertex set Wsatisfying [W ]g = V ([W ]m = V ).20th EWCG Seville, Spain (2004)20th European Workshop on Computational GeometryFor a nonempty set W ⊆ V , a connected sub-graph of G with the minimum number of edgesthat contains all of W must be clearly a tree. Sucha tree is called a Steiner W -tree. The Steiner in-terval S(W ) of W consists of all vertices that lieon some Steiner W -tree. If S(W ) = V , then W iscalled a Steiner set for G [3].The monophonic (m-hull, geodetic, g-hull, Stei-ner, respectively) number of G, denoted by mn(G)(mhn(G), gn(G), ghn(G), st(G), respectively) isthe minimum cardinality of a monophonic (m-hull,geodetic, g-hull, Steiner, respectively) set in G.Clearly, ghn(G) ≤ gn(G), since every geodetic setis a g-hull set. In [7] the authors showed that, apartfrom the previous one, no other general relation-ship among the parameters ghn(G), gn(G) andst(G) exists. In Section 2, we approach the sameproblem by replacing the parameter st(G) by bothmhn(G) and mn(G).In Section 3, we examine a number of mono-phonic convexity issues involving three types ofvertices: contour, peripheral and extreme ver-tices [2]. We prove, among other facts, that thecontour of a graph is monophonic. It is interest-ing to notice that this kind of results are closelyrelated to the graph reconstruction problem, inthe sense that we want to obtain all the verticesof a graph by considering a certain kind of pathsjoining vertices of a fixed set W .In [3], it was shown that every Steiner set in Gis also geodetic. Unfortunately, this particular re-sult turned out to be wrong and was disproved byPelayo [10]. In [7], the authors proved that everySteiner set is monophonic. As a consequence, theyimmediately derived that, in the class of distance-hereditary graphs (i.e., those graphs for whichevery monophonic path is a geodesic [8]), everySteiner set is geodetic. They also approached theproblem of determining for which classes of chordalgraphs (i.e., without induced cycles of lengthgreater than 3) every Steiner set is geodetic, prov-ing this statement to be true both for Ptolemaicgraphs (i.e, distance-hereditary chordal graphs [5])and interval graphs (i.e., chordal graphs withoutinduced asteroid triples [9]). In Section 4, we focusour attention on the edges of geodesic and mono-phonic paths, approaching the same problems andobtaining similar results.2. Monophonic and geodetic parametersLet us review the main definitions involved inthis section. A vertex set W ⊆ V is a g-hull setif its g-convex hull [W ]g covers all the graph, i.e.,if [W ]g = V . Moreover, W is called geodetic ifI[W ] = V . The g-hull number ghn(G) of G is de-fined as the minimum cardinality of a hull set. Thegeodetic number gn(G) of G is the minimum car-dinality of a geodetic set [6]. Certainly, ghn(G) ≤gn(G).Although it has been shown that determiningthe geodetic number and the hull number of agraph is a NP -hard problem [6], it is rather simpleto obtain these two parameters for a wide range ofclasses of graphs as paths, cycles, trees, (bipartite)complete graphs, wheels and hypercubes.A vertex set W ⊆ V is a m-hull set if [W ]m = V .Moreover, W is called monophonic if J [W ] = V .The m-hull number mhn(G) of G is the minimumcardinality of a m-hull set. The monophonic num-ber mn(G) of G is the minimum cardinality of amonophonic set. Certainly, mhn(G) ≤ mn(G) ≤gn(G) and mhn(G) ≤ ghn(G), since every mono-phonic set is a m-hull set, every geodetic set ismonophonic, and every g-hull set is a m-hull set.Nevertheless, it is not true that every g-hull set bemonophonic. For example, if we consider the com-plete bipartite graph K3,3, with V1 = {a, b, c} andV2 = {e, f, g}, it is easy to see that the set W ={a, b} satisfies [W ]g = V and J [W ] = V r {c}.At this point, what remains to be done is toask the following question: Is there any other gen-eral relationship among the parameters mhn(G),mn(G), ghn(G) and gn(G), apart from the previ-ous known inequalities?The next realization theorem shows that, unlesswe restrict ourselves to a specific class of graphs,the answer is negative.Theorem 1 For any integers a, b, c, d such that3 ≤ a ≤ b ≤ c ≤ d, there exists a connected graphG = (V,E) satisfying one of the following condi-tions:(i) a = mhn(G), b = mn(G), c = ghn(G) andd = gn(G),(ii) a = mnh(G), b = ghn(G), c = mn(G) andd = gn(G).March 25-26, 2004 Seville (Spain)3. Contour, peripheral and extremeverticesGiven a connected graph G = (V,E), the eccen-tricity of a vertex u ∈ V is defined as eccG(u) =ecc(u) = max{d(u, v)|v ∈ V }. Hence, the diameterD of G can be defined as the maximum eccentricityof the vertices in G. The periphery of G, denotedPer(G), is the set of vertices that have maximumeccentricity, i.e., the set of the so-called peripheralvertices. A vertex v is said to be simplicial in G ifthe subgraph induced by its neighborhood N(v) isa clique. The extreme set of G, denoted Ext(G), isthe set of all its simplicial vertices. With the aimof generalizing these two definitions, the so-calledcontour of G was introduced in [2] as follows. Avertex u ∈ W is said to be a contour vertex of Wif eccW(u) ≥ eccW(v), for all v ∈ N(u) ∩ W . Thecontour Ct(W ) of W is the set formed by all thecontour vertices of W . If W = V , this set is calledthe contour of G and it is denoted Ct(G). Noticethat Per(G) ∪ Ext(G) ⊆ Ct(G).Remark. We have examples of graphs showingthat there is no general relationship between pe-ripheral and extreme vertices.Let W ⊆ V be a m-convex (g-convex) set andlet F = 〈W 〉Gbe the subgraph of G induced by W .A vertex v ∈ W is called an m-extreme vertex (g-extreme vertex ) of W if W r {v} is a m-convex (g-convex) set. A vertex v of a m-convex (g-convex)set W is a m-extreme (g-extreme) vertex of W ifand only if v is simplicial in F [5].A convexity space (V, C) is a convex geometry ifit satisfies the so-called Minkowsky-Krein-Milmanproperty: Every convex set is the convex hull of itsextreme vertices. Notice that this condition allowsus to rebuild every convex set from its extreme ver-tices, by using the convex hull operator. Farber andJamison [5] proved that the monophonic (geodesic)convexity of a graph G is a convex geometry if andonly if G is chordal (Ptolemaic). Ca´ceres et al. [2]obtained a similar property to the previous one,valid for every graph, by considering, instead of theextreme vertices, the contour vertices.Theorem 2 [2] Let G = (V,E) be a connectedgraph and W ⊆ V a g-convex set. Then, W is theg-convex hull of its contour vertices.As was pointed out in [2], the contour of a graphneeds not to be geodetic. Nevertheless, we provethis assertion to be true in the following case.Proposition 3 If Ct(G) = Per(G), then Ct(G)is a geodetic set.PROOF. Let x be a vertex of V (G) r Ct(G).Since the eccentricities of two adjacent vertices dif-fer by at most one unit, if x is not a contour ver-tex, then there exists a vertex y ∈ V , adjacentto x, such that its eccentricity satisfies ecc(y) =ecc(x)+1. This fact implies the existence of a pathρ(x)= x0x1x2 . . . xr, such that x = x0, xi /∈ Ct(G)for i ∈ {0, . . . , r − 1}, xr ∈ Ct(G), and ecc(xi) =ecc(xi−1) + 1 = l + i for i ∈ {1, . . . , r}, where l =ecc(x). Moreover, ρ(x) is a shortest x − xr path,since otherwise, the eccentricity of xr would be lessthan l + r. But xr ∈ Ct(G) = Per(G) implies thatecc(xr) = D and D = l + r. Thus, there existsa vertex z ∈ Per(G) such that D = d(z, xr) ≤d(z, x)+d(x, xr) ≤ ecc(x)+r = l+r = D, that is,d(z, xr) = d(z, x)+d(x, xr). Hence, x is on a short-est path between the vertices z, xr ∈ Per(G) =Ct(G).As a consequence of this result, we have the fol-lowing corollary.Corollary 4 If Ct(G) has exactly two vertices,Ct(G) is a geodetic set.Now, we approach the same issues by consideringthe monophonic convexity.Theorem 5 The contour of any connected graphG is a monophonic set.Corollary 6 Let G be a connected graph and letW ⊆ V be a m-convex set. Then, every vertex of Wlies on a monophonic path joining contour verticesof W .PROOF. Let F = 〈W 〉Gbe the subgraph of Ginduced by W , and let Ω be the set of contourvertices of W . Certainly, Ω = Ct(F ). Hence, theprevious statement is equivalent to saying that thecontour of F is a monophonic set.Corollary 7 Let G be a connected graph and letW ⊆ V be a m-convex set. Then, W is the m-convex hull of its contour vertices.Finally, we show another consequence of Theo-rem 5, which was directly proved in [2].Corollary 8 The contour of a distance-hereditarygraph is a geodetic set.20th European Workshop on Computational Geometry4. The edge Steiner problemIn this section, we focus our attention on theedges that lie in paths joining two vertices of G =(V,E). We define the edge intervals of a graph asfollows. The edge geodetic closed interval Ie[u, v] isthe set of edges of all u − v geodesics. Similarly,the edge monophonic closed interval Je[u, v] is theset of vertices of all monophonic u − v paths. ForW ⊆ V , the edge geodetic closure Ie[W ] of W isthe union of all edge closed intervals Ie[u, v] overall pairs u, v ∈ W . The edge monophonic closure,Je[W ], is defined as the union of all edge closedmonophonic intervals over all pairs u, v ∈ W . Inother words, we haveIe[W ] =⋃u,v∈WIe[u, v], Je[W ] =⋃u,v∈WJe[u, v].A vertex set W for which Je[W ] = E is calledan edge monophonic set. Similarly, W is an edgegeodetic set if Ie[W ] = E [1]. A set W ⊆ V is anedge Steiner set if the edges lying in some SteinerW -tree cover E. Notice that: (1) every edge Steinerset is a Steiner set, (2) every edge geodetic set isgeodetic, (3) every edge monophonic set is mono-phonic, and (4) every edge geodetic set is an edgemonophonic set. It is easy to find examples wherethe converses of these statements are not true. Wehave obtained the following results.Theorem 9 Every edge Steiner set of a connectedgraph is an edge monophonic set.Corollary 10 In the class of connected distance-hereditary graph, every edge Steiner set is an edgegeodetic set.Analogously to the vertex case, this last resultalso holds for interval graphs. First, we need toprove the following lemma.Lemma 11 (i) If P is a x−y walk in G, any vertexof W is adjacent to at least a vertex of V (P ). (ii) IfTW is a Steiner W -tree, for any u ∈ V (TW )rW ,u lies in the unique x − y path of TW . Moreover,there exists a Steiner W -tree, T ∗W , formed by theunique x−y path of TW and vertices of W adjacentto vertices of that path.Theorem 12 In the class of connected intervalgraphs, every edge Steiner set is an edge geodeticset.In the preceding section we have seen that thecontour of a graph is a monophonic set. Neverthe-less, it is quite easy to find graphs whose contouris not an edge monophonic set.It remains an open question the problem of char-acterizing those classes of chordal graphs for whichevery edge Steiner set is edge geodetic. We knowthis statement to be true for interval and Ptolemaicgraphs, and false for split graphs (i.e., those chordalgraphs whose complementary is also chordal).References[1] M. Atici and A. Vince, Geodesics in graphs, anextremal set problem and perfect Hash families, Graphsand Combinatorics, 18, (3), 2002, pp. 403–413.[2] J. Ca´ceres, A. Ma´rquez, O. R. Oellerman, M. L.Puertas, Rebuilding convex sets in graphs, 19thEuropean Workshop on Computational Geometry,Bonn, 2003, submitted to Discrete Mathemathics.[3] G. Chartrand, P. Zhang, The Steiner number of agraph, Disc. Math., 242, 2002, pp. 41–54.[4] P. Duchet, Convex sets in graphs II. Minimal pathconvexity, J. Comb. Theory ser. B, 44, 1988, pp. 307–316.[5] M. Farber, R. E. Jamison, Convexity in graphs andhypergraphs, SIAM J. Alg. Disc. Math., 7 (3), 1986,pp. 433–444.[6] F. Harary, E. Loukakis, C. Tsouros, The geodeticnumber of a graph, Math. Comput. Modelling, 17 (11),1993, pp. 89–95.[7] C. Hernando, T. Jiang, M. Mora, I. M. Pelayo, C.Seara, On the Steiner, geodetic and hull numbersof graphs, 19th British Combinatorial Conference,Bangor 2003; submitted to Discrete Mathematics.[8] E. Howorka, A characterization of distance-hereditarygraphs, Quart. J. Math. Oxford Ser. 2, 28 (112), 1977,pp. 417–420.[9] C. G. Lekkerkerker, J. Ch. Boland, Representation ofa finite graph by a set of intervals on the real line,Fund. Math., 51, 1962/1963, pp. 45–64.[10] I. M. Pelayo, Not every Steiner set is geodetic, a noteon “The Steiner number of a graph”, Disc. Math.,accepted.[11] V. P. Soltan, Metric convexity in graphs, Studia Univ.Babes¸-Bolyai Math., 36, (4), 1991, pp. 3–43.

On geodesic and monophonic convexity

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On geodesic and monophonic convexity

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