On the Steiner, geodetic and hull numbers of graphs

Given a graph G and a subset W ? V (G), a Steiner W-tree is a tree of minimum order that contains all of W. Let S(W) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W) the Steiner interval of W. If S(W) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a shortest u − v path in G is called a u − v geodesic. Let I[u, v] denote the set of all vertices in G lying on some u − v geodesic, and let J[u, v] denote the set of all vertices in G lying on some induced u − v path. Given a set S ? V (G), let I[S] = ?u,v?S I[u, v], and let J[S] = ?u,v?S J[u, v]. We call I[S] the geodetic closure of S and J[S] the monophonic closure of S. If I[S] = V (G), then S is called a geodetic set of G. If J[S] = V (G), then S is called a monophonic set of G. The minimum order of a geodetic set in G is named the geodetic number of G. In this paper, we explore the relationships both between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. We thoroughly study the relationship between the Steiner number and the geodetic number, and address the questions: in a graph G when must every Steiner set also be geodetic and when must every Steiner set also be monophonic. In particular, among others we show that every Steiner set in a connected graph G must also be monophonic, and that every Steiner set in a connected interval graph H must be geodetic

On geodesic and monophonic convexity

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

Rebuilding convex sets in graphs

The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighborhoods are complete graphs. It is known that the class of graphs with the Minkowski–Krein–Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbor is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set

THE RESTRAINED STEINER NUMBER OF A GRAPH

For a connected graph G = (V, E) of order p, a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets. A set W of vertices of a graph G is a restrained Steiner set if W is a Steiner set, and if either W = V or the subgraph G[V − W ] induced by V − W has no isolated vertices. The minimum cardinality of a restrained Steiner set of G is the restrained Steiner number of G, and is denoted by s r (G). The restrained Steiner number of certain classes of graphs are determined. Connected graphs of order p with restrained Steiner number 2 are characterized. Various necessary conditions for the restrained Steiner number of a graph to be p are given. It is shown that, for integers a, b and p with 4 ≤ a ≤ b ≤ p, there exists a connected graph G of order p such that s(G) = a and s r (G) = b. It is also proved that for every pair of integers a, b with a ≥ 3 and b ≥ 3, there exists a connected graph G with s r (G) = a and g r (G) = b

Structure and properties of maximal outerplanar graphs.

Outerplanar graphs are planar graphs that have a plane embedding in which each vertex lies on the boundary of the exterior region. An outerplanar graph is maximal outerplanar if the graph obtained by adding an edge is not outerplanar. Maximal outerplanar graphs are also known as triangulations of polygons. The spine of a maximal outerplanar graph G is the dual graph of G without the vertex that corresponds to the exterior region. In this thesis we study metric properties involving geodesic intervals, geodetic sets, Steiner sets, different concepts of boundary, and also relationships between the independence numbers and domination numbers of maximal outerplanar graphs and their spines. In Chapter 2 we find an extension of a result by Beyer, et al. [3] that deals with Hamiltonian degree sequences in maximal outerplanar graphs. In Chapters 3 and 4 we give sharp bounds relating the independence number and domination number, respectively, of a maximal outerplanar graph to those of its spine. In Chapter 5 we discuss the boundary, contour, eccentricity, periphery, and extreme set of a graph. We give a characterization of the boundary of maximal outerplanar graphs that involves the degrees of vertices. We find properties that characterize the contour of a maximal outerplanar graph. The other main result of this chapter gives characterizations of graphs induced by the contour and by the periphery of a maximal outerplanar graph. In Chapter 6 we show that the generalized intervals in a maximal outerplanar graph are convex. We use this result to characterize geodetic sets in maximal outerplanar graphs. We show that every Steiner set in a maximal outerplanar graph is a geodetic set and also show some differences between these types of sets. We present sharp bounds for geodetic numbers and Steiner numbers of maximal outerplanar graphs