Mostafa Blidia

On b-colorings in regular graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. El-Sahili and Kouider have conjectured that every d-regular graph with girth at least 5 has a b-coloring with d+1 colors. We show that the Petersen graph infirms this conjecture, and we propose a new formulation of this question and give a positive answer for small degree.

Characterizations of trees with equal paired and double domination numbers

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.

A CHARACTERIZATION OF LOCATING-TOTAL DOMINATION EDGE CRITICAL GRAPHS

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number t(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) n D, NG(u)\ D 6 NG(v)\ D. The locating-total domination number t L(G)

Some operations preserving the existence of kernels

Abstract We consider some classical constructions of graphs: join of two graphs, duplication of a vertex, and show that they behave nicely from the point of view of the existence of kernels.

On kernels in perfect graphs

A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.

MAXIMAL k-INDEPENDENT SETS IN GRAPHS

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted ik(G) and k(G). We give some relations between k(G) and j(G) and between ik(G) and ij(G) for j 6 k. We study two families of extremal graphs for the inequality i2(G) i(G) + (G). Finally we give an upper bound on i2(G) and a lower bound when G is a cactus.

Ratios of Some Domination Parameters in Graphs and Claw-free Graphs

In the class of all graphs and the class of claw-free graphs, we give exact bounds on all the ratios of two graph parameters among the domination number, the total domination number, the paired domination number, the double domination number and the independence number. We summarize the old and new results in a table and give for each bound examples of extremal families.

On edge-b-critical graphs

A b-coloring of a graph G is a proper coloring of the vertices of G such that in each color class there exists a vertex having neighbors in all the other color classes. The b-chromatic number b ( G ) of a graph G is the largest integer such that G admits a b-coloring with k colors. A graph G is edge b-critical if deleting any edge decreases its b-chromatic number. We characterize the class of P 5 -free graphs, d -regular graphs and trees which are edge b-critical. As a consequence we show that deciding if a graph is edge-b-critical is NP-hard.

On the p-domination number of cactus graphs

Let p be a positive integer and G = (V, E) a graph. A subset S of V is a p-dominating set if every vertex of V − S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γp(G). It is proved for a cactus graph G that γp(G) 6 (|V |+ |Lp(G)|+c(G))/2, for every positive integer p > 2, where Lp(G) is the set of vertices of G of degree at most p − 1 and c(G) is the number of odd cycles in G.

A characterization of edge b-critical graphs

Abstract Let G be a simple graph. A b-coloring of G is a proper coloring of its vertices such that in each color class there exists a vertex having neighbors in all the other color classes. The b-chromatic number of G , denoted by b ( G ) , is the maximum number k such that G admits a b-coloring with k colors. A graph G is called edge b-critical if deleting any edge decreases its b-chromatic number. Our main purpose is to characterize all such graphs.