For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n . In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n ) = ω(G ...

Space partitioning techniques are well known especially because of their use in computer graphics, especially within ray-tracing acceleration techniques. The primary aim of those techniques is to enable fast test whether a geometric object resides at least partially within the given area. There are many modifications that proved the applicability despite of high memory requirements of complexit...

We show that for every minimum eternal dominating set, D, of a graph G and every vertex v ∈ D, there is a sequence of attacks at the vertices of G which can be defended in such a way that an eternal dominating set not containing v is reached. The study of the stronger assertion that such a set can be reached after a single attack is defended leads to the study of graphs which are critical in th...

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...

The eternal vertex cover problem is a variant of the classical defined in terms an infinite attacker–defender game played on graph. In each round game, defender reconfigures guards from one to another response move by attacker. minimum number required any winning strategy when this graph G , denoted evc ( ) . It known that given and integer k checking whether ≤ NP-hard. Further, it for mvc 2 wh...

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can “move” to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence...