M. Carmen Hernando

On the metric dimension of some families of graphs

Abstract The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [G. Chartrand, D. Erwin, G. L. Johns and P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34; C. Poisson and P. Zhang, The metric dimension of unicyclic graphs, J. Comb. Math Comb. Comput. 40 (2002) 17–32], Robotic Navigation [S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Disc. Appl. Math. 70 (1996) 217–229; B. Shanmukha, B. Sooryanarayana and K. S. Harinath, Metric dimension of wheels, Far East J. Appl. Math. 8 (3) (2002) 217–229] and Combinatorial Search and Optimization [A. Sebo and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2) (2004) 383–393]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study its behavior with respect to the join and the cartesian product of graphs.

On the geodetic and the hull numbers in strong product graphs

A set S of vertices of a connected graph G is convex, if for any pair of vertices u,v@?S, every shortest path joining u and v is contained in S. The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the minimum cardinality of a geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also establish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.

Graphs of non-crossing perfect matchings

Abstract. Let Pn be a set of n=2m points that are the vertices of a convex polygon, and let ℳm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2=M1−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of Pn. We prove the following results about ℳm: its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4.

On local transformation of polygons with visibility properties

One strategy for the enumeration of a class of objects is local transformation, in whch new objects of the class are produced by means of a small modification of a previously-visited object in the same class. When local transformation is possible, the operation can be used to generate objects of the class via random walks, and as the basis for such optimization heuristics as simulated annealing.For general simple polygons on fixed point sets, it is still not known whether the class of polygons on the set is connected via a constant-size local transformation. In this paper, we exhibit a simple local transformation for which the following polygon classes are connected: monotone, x-monotone, star-shaped, (weakly) edge-visible and (weakly) externally visible. The latter class is particularly interesting as it is the most general polygon class known to be connected under local transformation. For each of the polygon classes, we also provide asymptotically-tight worst-case upper bounds on the minimum number of operations required to transform one member of the class to any other.

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.

Nordhaus-Gaddum bounds for locating domination

A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating-dominating. Locating, metric-locating-dominating and locating-dominating sets of minimum cardinality are called @b-codes, @h-codes and @l-codes, respectively. A Nordhaus-Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement G@?. In this paper, we present some Nordhaus-Gaddum bounds for the location number @b, the metric-location-domination number @h and the location-domination number @l. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.

Locating-dominating codes: Bounds and extremal cardinalities

In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our contributions on @l-codes and @h-codes concerning bounds, extremal values and realization theorems.

Searching for geodetic boundary vertex sets

Av ertexv is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices, including a realization theorem that not only corrects a wrong statement detected in [2], but also improves it. We also prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, every vertex in G lies on some shortest path joining two boundary vertices. A vertex v belongs to the contour Ct(G )o fG if no neighbor of v has an eccentricity greater than those of v. We study the geodeticity of the contour Ct(G) and other related sets.

On the metric dimension of infinite graphs

A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.

Some structural, metric and convex properties on the boundary of a graph

Abstract Let u , v ∈ V be two vertices of a connected graph G. The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v. The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under different points of view: (1) a realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary; (2) the boundary is an edge-geodetic set, and the contour is a monophonic set; (3) the boundary is a resolving set.