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 graph G is called Bk-VPG (resp., Bk-EPG), for some constant k ≥ 0, if it has a string representation on a grid such that each vertex is an orthogonal path with at most k bends and two vertices are adjacent in G if and only if the corresponding strings intersect (resp., the corresponding strings share at least one grid edge). If two adjacent strings of a Bk-VPG graph intersect exactly once, th...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...

Given a graph G = (V,E), a perfect dominating set is a subset of vertices V ′ ⊆ V (G) such that each vertex v ∈ V (G) \ V ′ is dominated by exactly one vertex v ∈ V . An efficient dominating set is a perfect dominating set V ′ where V ′ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remain...

A subset S of vertices in a graph G is a called a geodetic dominating set if S is both a geodetic set and a (standard) dominating set. In this paper, we study geodetic domination on graphs.

In this paper, we study the Dominating Set problem with measure functions, which is extended from the general Dominating Set problem. We study the correspondnig problems on complexity, approximation and inapproximability for Dominating Set problem with measure functions. In addition, we extend our results to the weighted graphs.

Preprocessing by data reduction is a simple but powerful technique used for practically solving different network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads efficiently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of gr...

In this paper we study strongest paths in a fuzzy graph. A necessary and sufficient condition for an arc in a fuzzy graph to be a strongest path and a sufficient condition for a path in a fuzzy graph to be a strongest path are obtained. A characterization of δ−arcs and the relationship between fuzzy cutnodes and δ−arcs are also obtained. Also a characterization of blocks is obtained using stron...