Set domination in graphs

Abstract

Let G = (V, E ) be a connected graph. A set D C V is a set-dominating set (sd-set) if for every set T C V -D, there exists a nonempty set S C Dsuch that the subgraph (S U T ) induced by S U Tis connected. The set-domination number ys(G) of G is the minimum cardinality of a sd-set. In this paper we develop properties of this new parameter and relate it to some other known domination parameters. A set D C V is a set-dominating set (sd-set) of G if for every set T C V -D , there exists a nonempty subset S C D such that the subgraph (S U T) is connected. The set-domination number y,(G) of G is the minimum cardinality of a sd-set.Besides obtaining some bounds for y,(G), we relate it to some other known parameters of G.

The Global Set-Domination Number in Jump Graphs

“…A set D V(J(G) is a dominating set of J(G) if every vertex not in D is adjacent to some vertex in D. Further d is a global dominating set of J(G), if D is a dominating set of both J(G) and J( ) . The domination number √(J(G)) of J(G) is defined similarly the concept of global domination was first introduced by sampathkumar [4] and was also studied by Rall [3] Recently the concept of set domination for a connected graph was introduced by Sampath kumar and L. pushpa latha [ 5]. A set D V(J(G) is an set-dominating set (sd-set)of every set S V(J(G))-D, there existd a non empty set T D such that the sub graph < S T > induced by S Tis connected.…”

“…In a jump tree J(T) with p vertices and e end vertices that is not a star the set of non-end vertices form a minimum global sd-set and √ sg J(T) =p-e.Proof: It is known that the set d of all cut vertices of T form a √ s -set of T and √ s = pe[5] Clearly the sub graph V(J(T))-D in J( ) is complete. Since J(T) ≠ K 1,m in J( ) each vertex in V(J(T)) -D is adjacent to some vertex in D this implies that D is an sd-set of J(T) also and √ sg = peWe now determine some bounds for √ sg .…”

About ve-Domination in Graphs

“…In [3], Sampathkumar and others have mentioned that for a triangle free graph the concepts of vertex covering and ve-domination are the same.…”

“…A set S of vertices is said to be a vedominating set if every edge of the graph G is m-dominated by some vertex in S . This concept is well studied in [3].…”

Co-Secure Set Domination in Graphs

“…which is minimum cardinality of a dominating set of G [4], [5]. For any connected graph G, a subset D of V(G) is a set dominating set of G if for every M⊆V/D there exists a non-empty set N⊆D such that the sub graph <MUN> induced by MUN is connected and its minimum cardinality of a set dominating set of G is the set domination number and is denoted by   G γ s and is abbreviated as a SD-set (set dominating set) [6]. For any non-trivial graphs G with a subset D of the vertex set V(G) is a co-secure dominating set of a graph G if D is a dominating set and for every…”