The distance dG(u, v) between two vertices u and v in a connected graph G is the length of the shortest uv-path in G. A uv-path of length dG(u, v) is called uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for every two vertices a, b ∈ X. The convex domination number γcon(G) of a graph G equals the minimum cardinality of a convex dominating set. There are a larg...

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

Let G = (V,E) be a simple graph. A subset Dof V (G) is a (k, r)dominating set if for every vertexv ∈ V −D, there exists at least k vertices in D which are at a distance utmost r from v in [1]. The minimum cardinality of a (k, r)-dominating set of G is called the (k, r)-domination number of G and is denoted by γ(k,r)(G). In this paper, minimal (k, r)dominating sets are characterized. It is prove...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

Let G= (V ,E) be a digraph with a distinguished set of terminal vertices K ⊆ V and a vertex s ∈ K . We define the s,K-diameter of G as the maximum distance between s and any of the vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained s,K-terminal reliability of G, Rs,K(G,D), is defined as the probability ...

A unit disk graph is the intersection graph of a set of unit diameter disks in the plane. In this paper we consider liar’s domination problem on unit disk graphs, a variant of dominating set problem. We call this problem as Euclidean liar’s domination problem. In the Euclidean liar’s domination problem, a set P = {p1, p2, . . . , pn} of n points (disk centers) are given in the Euclidean plane. ...

The TAG adjunction operation operates by splitting a tree at one node, which we will call the adjunction site. In the resulting structure, the sub-trees above and below the adjunction site are separated by, and connected with, the auxiliary tree used in the composition. As the adjunction site is thus split into two nodes, with a copy in each subtree, a natural way of formalizing the adjunction ...

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...