Joël Puech

Total domination in graphs with minimum degree three

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.

Decomposable trees: a polynomial algorithm for tripodes

In this article, we deal with graphs modelling interconnection networks of parallel systems (parallel computers, networks of workstations, etc.). We want to share the nodes of such a network between many users, each one needing a given number of nodes. Thus, a graph G with N vertices is said to be decomposable if for each set {n1,..., nk} whose sum is equal to N, there exists a partition V1,...,Vk of V(G) such that for each i, 1 ≤i≤k, |Vi| = ni and the subgraph induced by Vi is connected. We show that determining whether a given tripode (three disjoint chains connected by one extremity to a same new vertex) is decomposable can be done by a polynomial algorithm.

A characterization of (γ, i)-trees

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues.

Irredundance perfect and P 6 -free graphs

Let G be a connected graph on n vertices. A spanning tree T of G is called an independence tree, if the set of end vertices of T (vertices with degree one in T) is an independent set in G. If G has an independence tree, then α t(G) denotes the maximum number of end vertices of an independence tree of G. We show that determining αt of a graph is an NP-hard problem. We give the following analogue of a well-known result due to Chvatal and Erdos. If αt(G) ≤ κ(G) - 1, then G is hamiltonian. We show that the condition is sharp. An I≤k-tree of G is an independence tree of G with at most k end vertices or a Hamilton cycle of G. We prove the following two generalizations of a theorem of Ore. If G has an independence tree T with k end vertices such that two end vertices of T have degree sum at least n - k + 2 in G, then G has an I≤k-1-tree. If the degree sum of each pair of nonadjacent vertices of G is at least n - k + 1, then G has an I≤k-tree. Finally, we prove the following analogue of a closure theorem due to Bondy and Chvatal. If the degree sum of two nonadjacent vertices u and v of G is at least n - 1, then G has an I≤k-tree if and only if G + uv has an I≤k-tree (k ≥ 2). The last three results are all best possible with respect to the degree sum conditions.

Irredundance and domination in kings graphs

Each king on an n × n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n - 1)2/3 and n2/3, and that the minimum size of a maximal irredundant set of kings is bounded between n2/9 and ⌈(n + 2)/3⌉2, where the latter upper and lower bounds are in fact equal when n ≡ 0(mod3). Results are given for related domination and independence problems.

On domination and annihilation in graphs with claw-free blocks

Let � (G), ra(G) and ir(G) denote the domination, R-annihilation and irredundance numbers of a graph G, respectively. Graphs whose blocks are claw-free are called CFB-graph. In this paper we establish the best possible upper bounds on the ratios � (G)=ra(G) and � (G)=ir(G )i n the class of CFB-graphs. The CFB-graphs generalize several classes of graphs for which such ratios have already been investigated. Motivated by our proof methods, we are led to introduce a new family of domination parameters simultaneously generalizing the total domination and k-domination numbers. For two integers, l?0 and k? 0 a set X of vertices of a graph G=(V; E) is an l-total k-dominating set of G, if every vertex in X has at least l neighbors in X and every vertex in V \ X has at least k neighbors in X . If (at least) one l-total k-dominating set exists,

An inequality chain of domination parameters for trees

We prove that the smallest cardinality of a maximal packing in any tree is at most the cardinality of an R-annihilated set. As a corollary to this result we point out that a set of parameters of trees involving packing, perfect neighbourhood, R-annihilated, irredundant and dominating sets is totally ordered. The class of trees for which all these parameters are equal is described and we give an example of a tree in which most of them are distinct.

R -annihilated and independent perfect neighborhood sets in chordal graphs

Abstract Let θ i ( G ), ra( G ) be the minimum cardinality of, respectively, an independent perfect neighborhood set and an R-annihilated set. We point out some classes of graphs for which the inequality θ i ( G )⩽ra( G ) holds. This study is natural since Favaron and Puech (Discrete Math. 197/198 (1999) 269–284) contains examples of graphs where the difference θ i ( G )−ra( G ) is positive and can be arbitrarily large. We prove that the inequality θ i ( G )⩽ra( G ) holds if the cycles of G satisfy some assertions verified in particular by the chordal graphs. This result generalizes the one concerning trees proved in Cockayne et al. (Discrete Math. 188 (1948) 253–260). We also establish the same inequality for C 1,2,2 -free graphs, which generalizes the result proved independently in Cockayne and Mynhardt (J. Combin. Math. Combin. Comput., to appear) and Favaron and Puech for claw-free graphs.