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

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.

On Graphs with Prescribed Clique Number and Point-Arboricity

Journal of the London Mathematical Society, 1971

The chromatic number of a graph G may be defined as the minimum number of subsets in any partition of the point set of G so that each subset induces a subgraph with no lines. By replacing " lines " with " cycles " in the preceding statement we obtain the definition of point-arboricity. More formally, the point-arboricity p(G) of a graph G is the minimum number of subsets in any partition of the point set of G so that each subset induces an acyclic subgraph. This parameter was considered by Motzkin [5] and was investigated in . The clique number o)(G) of a graph G is the maximum number of points in any complete subgraph of G. Zykov proved that for any integers k and d where 2 ^ d < k, there exists a graph G with chromatic number k such that (o(G) = d. The purpose of this paper is to present an analogue to Zykov's theorem for pointarboricity. In order to do this, it is convenient to define some terms and establish some notation which are used throughout the paper. Denote the point set and line set of graph G by V(G) and has no cycles and is independent if (W} is totally disconnected. Furthermore, [x] and {x} denote the greatest integer not exceeding x and the least integer not less than x, respectively. We first prove a special case of the main result. THEOREM 1. For each positive integer m there exists a graph G m with no triangles such that p(G m ) = m.

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.

On clique convergent graphs

Graphs and Combinatorics, 1995

A graph G is convergent when there is some nite integer n 0, such that the n-th iterated clique graph K n (G) has only one vertex. The smallest such n is the index of G. The Helly defect of a convergent graph is the smallest n such that K n (G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the di erence of its index and diameter by more than one. In the present paper an a rmative answer to the question is given. For any arbitrary nite integer n, a graph is exhibited, the Helly defect of which exceeds by n the di erence of its index and diameter.

On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs

Discrete Applied Mathematics, 2010

On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs* * 1. Introduction A complete set of a graph G = (V. E) is a subset of V inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph K(G) of G is the intersection graph of C(G). Say that G is a clique graph if there exists a graph H such that G = K(H). A clique-Helly graph is a graph where C(G) satisfies the Helly property. any pairwise intersecting subfamily of C(G) has non-empty total intersection [1]. A hereditary clique-Helly graph is a graph where every induced subgraph is clique-Helly. The class of hereditary clique-Helly graphs (hXH) is contained in the class of clique-Helly graphs (XH), which in turn is contained in the class of clique graphs (X). Clique graphs and subclasses have been much studied as intersection graphs, in the context of graph operators, and are included in several books [2-5], An extended abstract of this paper was presented at LAGOS 2007, the IV Latin-American algorithms, graphs and optimization symposium, and appeared in Electronic Notes in Discrete Mathematics 30 (2008) 147-152. This research was partially supported by CNPq, Prosul-CNPq, CAPES (BrazilJ/COFECUB (France), Prociencia Project-FAPERJ.

Helly property, clique graphs, complementary graph classes, and sandwich problems

Journal of the Brazilian Computer Society, 2008

A sandwich problem for property Π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property Π. Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is N P-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all N P-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.

Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

2009

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem.

The Erdős–Pósa property for clique minors in highly connected graphs

Journal of Combinatorial Theory, Series B, 2012

We prove the existence of a function f : N 2 → N such that, for all p, k ∈ N, every (k(p − 3) + 14p + 14)-connected graph either has k disjoint Kp minors or contains a set of at most f (p, k) vertices whose deletion kills all its Kp minors. For fixed p ≥ 5, the connectivity bound of about k(p − 3) is smallest possible, up to an additive constant: if we assume less connectivity in terms of k, there will be no such function f .

Cliques in graphs

2010

The main focus of this thesis is to evaluate kr(n, δ), the minimal number of r-cliques in graphs with n vertices and minimum degree δ. A fundamental result in Graph Theory states that a triangle-free graph of order n has at most n/4 edges. Hence, a triangle-free graph has minimum degree at most n/2, so if k3(n, δ) = 0 then δ ≤ n/2. For n/2 ≤ δ ≤ 4n/5, I have evaluated kr(n, δ) and determined the structures of the extremal graphs. For δ ≥ 4n/5, I give a conjecture on kr(n, δ), as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let k r (n, δ) be the analogous version of kr(n, δ) for regular graphs. Notice that there exist n and δ such that kr(n, δ) = 0 but k reg r (n, δ) > 0. For example, a theorem of Andrasfai, Erdős and Sos states that any triangle-free graph of order n with minimum degree greater than 2n/5 must be bipartite. Hence k3(n, bn/2c) = 0 but k 3 (n, bn/2c) > 0 for n odd. I have evaluated ...

The Clique Transversal and Clique Independence of Distance Hereditary Graphs

2002

A clique-transversal set of a graph G is a sub- set of vertices intersecting all maximal cliques of G. The smallest possible cardinality τC (G) among all clique transversal sets for G is called clique transversal number. A clique independence set is a collection of vertex-disjoint maximal cliques of G. The largest possible cardinality αC (G) among all clique independent sets for G is called clique independence number. If equality of τC (H) and αC (H) always holds for every induced subgraph H of G, then the graph is called clique-perfect. The paper concentrates on the clique transver- sal and clique independence of distance-hereditary graphs. We present two polynomial time algo- rithms for finding the clique transversal number and the clique independence number of a given distance-hereditary graph.

On clique-transversals and clique-independent sets

Annals of Operations Research, 2002

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A cliqueindependent set is a subset of pairwise disjoint cliques of G. Denote by τ C (G) and α C (G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τ C (H ) = α C (H ), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τ C (G) and α C (G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τ C (G) − α C (G) is arbitrarily large.

Characterization and recognition of Helly circular-arc clique-perfect graphs

2005