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A Characterization of Trees for a New Lower Bound on the K-Independence Number

Nacéra MeddahMostafa Blidia — 2013

Discussiones Mathematicae Graph Theory

Let k be a positive integer and G = (V,E) a graph of order n. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we show that for every graph [xxx], where χ(G), s(G) and Lv are the chromatic number, the number of supports vertices and the number of leaves neighbors of v, in the graph G, respectively. Moreover,...

A characterization of locating-total domination edge critical graphs

Mostafa BlidiaWidad Dali — 2011

Discussiones Mathematicae Graph Theory

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 γₜ(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)∖D, N G ( u ) D N G ( v ) D . The locating-total domination number γ L t ( G ) is the minimum cardinality of a locating-total dominating set...

On the p-domination number of cactus graphs

Mostafa BlidiaMustapha ChellaliLutz Volkmann — 2005

Discussiones Mathematicae Graph Theory

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) ⩽ (|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.

Maximal k-independent sets in graphs

Mostafa BlidiaMustapha ChellaliOdile FavaronNacéra Meddah — 2008

Discussiones Mathematicae Graph Theory

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 βₖ(G). We give some relations between βₖ(G) and β j ( G ) and between iₖ(G) and i j ( G ) for j ≠ k. We study two families of extremal graphs for the inequality i₂(G) ≤ i(G) + β(G). Finally we give an upper bound on i₂(G) and a lower bound when G is a cactus.

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