A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. The matching number of G is the maximum cardinality of a matching of G. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. If G does not contain K1,...

The relationship ρL(G) ≤ ρ(G) ≤ γ (G) between the lower packing number ρL(G), the packing number ρ(G) and the domination number γ (G) of a graph G is well known. In this paper we establish best possible bounds on the ratios of the packing numbers of any (connected) graph to its six domination-related parameters (the lower and upper irredundance numbers ir and IR, the lower and upper independenc...

In this paper, we study the Goldberg Snarks Gk, twist Goldberg Snarks TGk, Flower snarks Fk, and show the exact value of connected and tree domination number of them. Mathematics Subject Classification: 05C35

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of t...

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in ...

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least 7. We show that the domination number of a connected graph with girth at least 7 is either 1 or at least 1 2 (3 + √ 8(m− n) + 9), where n is the number of vertices in the graph and m is the number of edges in the graph. For graphs with minimum degree 2 and girth at least 7, the lower bound c...

Let G = (V,E) be a connected graph. A connected dominating set S ⊂ V is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted γc(G), is the minimum cardinality of a connected dominating set. Alternatively, |V | − γc(G) is the maximum number of leaves in a spanning tree of G. Let δ denote the minimum degree of G. We prove that γc(G) ≤ |V | ln(δ+1)...