Changing and unchanging of the domination number of a graph: path addition numbers

Changing and unchanging of the domination number of a graph

LetG be a graph and (G) denote the domination number ofG. A dominating setD of a graphGwith |D|= (G) is called a -set ofG. A vertex x of a graphG is called: (i) -fixed if x belongs to every -set, (ii) -free if x belongs to some -set but not to all -sets, (iii) -bad if x belongs to no -set, (iv) −-free if x is -free and (G− x)= (G)− 1, (v) 0-free if x is -free and (G− x)= (G), and (vi) q -fixed ...

Changing and unchanging acyclic domination: edge addition

Changing and Unchanging Domination Number of a Commutative Ring

Let R be a commutative ring and let Γ(Zn) be the zero divisor graph of a commutative ring R, whose vertices are non-zero zero divisors of Zn, and such that the two vertices u, v are adjacent if n divides uv. In this paper, we evaluate the changing and unchanging domination number of Γ(Zn) and we evaluate the same result in Γ(Z2p), Γ(Z3p), Γ(Zp2), Γ(Z2n) and Γ(Z3n) . Our main result is, Brigham ...

Changing and Unchanging of Complementary Tree Domination Number in Graphs

A set D of a graph G = (V,E) is a dominating set if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. A dominating set D is called a complementary tree dominating set if the induced subgraph < V −D > is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domina...

The convex domination subdivision number of a graph

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...