Helly theorems for 3-Steiner and 3-monophonic convexity in graphs (original) (raw)

The Convexity Number of a Graph

Graphs and Combinatorics, 2002

For two vertices u and v of a connected graph G, the set I½u; v consists of all those vertices lying on a u À v shortest path in G, while for a set S of vertices of G, the set I½S is the union of all sets I½u; v for u; v 2 S. A set S is convex if I½S ¼ S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number xðGÞ is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n ! 3 and 2 xðGÞ conðGÞ n À 1. It is shown that for every triple l; k; n of integers with n ! 3 and 2 l k n À 1, there exists a noncomplete connected graph G of order n with xðGÞ ¼ l and conðGÞ ¼ k. Other results on convex numbers are also presented.

Characterization and recognition of generalized clique-Helly graphs

Discrete Applied Mathematics, 2007

Let p 1 and q 0 be integers. A family of sets F is (p, q)-intersecting when every subfamily F ⊆ F formed by p or less members has total intersection of cardinality at least q. A family of sets F is (p, q)-Helly when every (p, q)-intersecting subfamily F ⊆ F has total intersection of cardinality at least q. A graph G is a (p, q)-clique-Helly graph when its family of (maximal) cliques is (p, q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p = 2, q = 1. In this work we present a characterization for (p, q)-clique-Helly graphs. For fixed p, q, this characterization leads to a polynomial-time recognition algorithm. When p or q is not fixed, it is shown that the recognition of (p, q)-clique-Helly graphs is NP-hard.

A new proof of the Fisher-Ryan bounds for the number of cliques of a graph

2000

Tutte and Nash-Williams, independently, gave necessary and sufficient conditions for a connected graph to have at least t edgedisjoint spanning trees. Gusfield introduced the concept of edgetoughness η(G) of a connected graph G, defined as the minimum |S|/(ω(G − S) − 1) taken over all edge-disconnecting sets S of G, where ω(G − S) is the number of connected components of G − S. If a graph has edge-toughness η(G), Tutte and Nash-Williams's theorem says that the maximum number of edge-disjoint spanning trees of a graphs is given by η(G). Kundu used this result to show that a graph with edge-connectivity λ(G) has at least λ(G)/2 edgedisjoint spanning trees. In this paper we investigate to which extent the above results can be generalized to a graph G = (V, E) with a distinguished subset of vertices K. We obtain lower bounds for the maximum number of edge-disjoint Steiner trees of G (minimal trees of G containing K) in terms of λK (G), the K-edge-connectivity of G (defined as the minimum number of edges whose removal disconnects K). In [3] we introduced the K-edge-toughness of a graph, ηK (G) (which coincides with η(G) when K = V). We extent some of the properties of the edge-toughness of a graph to the K-edge-toughness and we show by mean of a counterexample that the maximum number of disjoint Steiner trees can be less than ηK (G) when K = V. We conclude with some conjectures regarding these bounds.

On the Convexity Number of Graphs

Graphs and Combinatorics, 2012

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G.

Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond

arXiv (Cornell University), 2015

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in permutation graphs and the geodetic number in P 4-sparse graphs.

On hereditary Helly classes of graphs

In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.

Complexity aspects of the Helly property: Graphs and hypergraphs

In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. A family of subsets has the Helly property when every subfamily thereof, formed by pairwise intersecting subsets, contains a common element. Many generalizations of this property exist which are relevant to some fields of mathematics, and have several applications in computer science. In this work, we survey complexity aspects of the Helly property. The main focus is on characterizations of several classes of graphs and hypergraphs related to the Helly property. We describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NP-hardness results.