we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

‎in this paper‎, ‎we define the common minimal dominating signed‎ ‎graph of a given signed graph and offer a structural‎ ‎characterization of common minimal dominating signed graphs‎. ‎in‎ ‎the sequel‎, ‎we also obtained switching equivalence‎ ‎characterizations‎: ‎$overline{s} sim cmd(s)$ and $cmd(s) sim‎ ‎n(s)$‎, ‎where $overline{s}$‎, ‎$cmd(s)$ and $n(s)$ are complementary‎ ‎signed gra...

in this paper, we define the common minimal common neighborhooddominating signed graph (or common minimal $cn$-dominating signedgraph) of a given signed graph and offer a structuralcharacterization of common minimal $cn$-dominating signed graphs.in the sequel, we also obtained switching equivalencecharacterization: $overline{sigma} sim cmcn(sigma)$, where$overline{sigma}$ and $cmcn(sigma)$ are ...

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...

Abstract. The middle edge dominating graph Med(G) of a graph G=(V,E) is a graph with the vertex set E ∪S where S is the set of all minimal edge dominating set G and with two vertices u, v є E ∪S adjacent if u є E and V=F is a minimal edge dominating set of G containing u or u,v are not disjoint minimal edge dominating sets in S .In this paper we discuss about the middle edge dominating graph of...

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bound...

The maximum number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.7159. This upper bound might not be tight, since no examples of graphs with 1.5705 or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the maximum number of minimal dominating sets in graphs on n vertices. In some cases, we ...

A vertex subset S in a graph G is a dominating set if every vertex not contained in S has a neighbor in S. A dominating set S is a connected dominating set if the subgraph G[S] induced by S is connected. A connected dominating set S is a minimal connected dominating set if no proper subset of S is also a connected dominating set. We prove that there exists a constant ǫ > 10 such that every grap...

In this paper we introduce the common minimal equitable and vertex minimal equitable dominating graph and we get characterize the common minimal equitable and vertex minimal equitable dominating graph which are either connected or complete, some new results of these graphs are obtained. Mathematics Subject Classification: 05C70