Mustapha Chellali

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.

On locating and differetiating-total domination in trees

A total dominating set of a graph G = (V, E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V − S, N(u) ∩ S 6= N(v) ∩ S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N [u]∩ S 6= N [v]∩ S. Let γ t (G) and γ t (G) be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with ` leaves and s support vertices, γ t (T ) > max{2(n+ `− s+1)/5, (n+2− s)/2}, and for a tree of order n ≥ 3, γ t (T ) ≥ 3(n+`−s+1)/7, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying γ t (T ) = 2(n + ` − s + 1)/5 or γ t (T ) = 3(n + ` − s + 1)/7.

[1, 2]-sets in graphs

A subset S@?V in a graph G=(V,E) is a [j,k]-set if, for every vertex v@?V@?S, j@?|N(v)@?S|@?k for non-negative integers j and k, that is, every vertex v@?V@?S is adjacent to at least j but not more than k vertices in S. In this paper, we focus on small j and k, and relate the concept of [j,k]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G, the restrained domination number is equal to the domination number of G.

Dominator Colorings in Some Classes of Graphs

A dominator coloring is a coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class. We present new bounds on the dominator coloring number of a graph, with applications to chordal graphs. We show how to compute the dominator coloring number in polynomial time for P4-free graphs, and we give a polynomial-time characterization of graphs with dominator coloring number at most 3.

Ratios of some domination parameters in trees

Abstract We determine upper bounds on the ratios of several domination parameters in trees.

Global alliances and independence in trees

A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V − S has at least one neighbor in S, and for each vertex v in S (respectively, in V − S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborhood of v must be in S. For each kind of alliance, the associated parameter is the minimum cardinality of such an alliance. We determine relationships among these four parameters and the vertex independence number for trees.

Bounds on weak roman and 2-rainbow domination numbers

Abstract We mainly study two related dominating functions, namely, the weak Roman and 2-rainbow dominating functions. We show that for all graphs, the weak Roman domination number is bounded above by the 2-rainbow domination number. We present bounds on the weak Roman domination number and the secure domination number in terms of the total domination number for specific families of graphs, and we show that the 2-rainbow domination number is bounded below by the total domination number for trees and for a subfamily of cactus graphs.

Strong Equality Between the Roman Domination and Independent Roman Domination Numbers in Trees

Abstract A Roman dominating function (RDF) on a graph G = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = P u2V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G), denoted by R(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with R(T) ≡ iR(T).

Roman { 2 } -domination

In this paper, we initiate the study of a variant of Roman dominating functions. For a graph G = ( V , E ) , a Roman { 2 } -dominating function f : V ? { 0 , 1 , 2 } has the property that for every vertex v ? V with f ( v ) = 0 , either v is adjacent to a vertex assigned 2 under f , or v is adjacent to least two vertices assigned 1 under f . The weight of a Roman { 2 } -dominating function is the sum ? v ? V f ( v ) , and the minimum weight of a Roman { 2 } -dominating function f is the Roman { 2 } -domination number. First, we present bounds relating the Roman { 2 } -domination number to some other domination parameters. In particular, we show that the Roman { 2 } -domination number is bounded above by the 2-rainbow domination number. Moreover, we prove that equality between these two parameters holds for trees and cactus graphs with no even cycles. Finally, we show that associated decision problem for Roman { 2 } -domination is NP-complete, even for bipartite graphs.

On the dominator colorings in trees

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number χd(G) is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies γ(T ) + 1 ≤ χd(T ) ≤ γ(T ) + 2. In this note we characterize nontrivial trees T attaining each bound.