Abstract. For a vertex v of a connected graph G(V, E) with vertex set V(G), edge set E(G) and S ⊆ V(G). Given an ordered partition Π = {S 1 , S 2 , S 3 , ..., S k } of the vertex set V of G, the representation of a vertex v ∈ V with respect to Π is the vector is a resolving partition if different vertices of G have distinct representations, i.e., for every pair of vertices u, v ∈ V(G), r(u|Π) r(v|Π). The minimum k of Π resolving partition is a partition dimension of G, denoted by pd(G). Finding the partition dimension of G is classified to be a NP-Hard problem. In this paper, we will show that the partition dimension of comb product of path and complete graph. The results show that comb product of complete grapph K m and path P n namely pd(K m P n ) = m where m ≥ 3 and n ≥ 2 and pd(P n K m ) = m where m ≥ 3, n ≥ 2 and m ≥ n.
Given a simple graph G. Vertex set of G is V, and edge set of G is E. Domination set, denoted by S, that is subset of V such that every vertex in V which is not element of S has distance one to S. The least number of the elements of S is the domination number of the graph G, that is ϒ(G). Let G1 and G2 be a simple graph. G1 has n1 vertices, and has m1 edges. G2 has n2 vertices, and has m2 edges. We defined an operator called neighbourhood corona, denoted by a star ‘*’. Graph G1*G2 is a new graph obtained by making ni copies of second graph and for each i make connecting all vertices in i-th copy of second graph G2 to neighbours of vi = 1, 2, …, n. Furthermore, new graph 2-neighbourhood corona G1*2G2, has n1 copies of G2 and for each i make connecting to all vertices of ith copy of G2 to neighbours of vi, i = 1, 2, …, n. In this research, we determined ϒ(Gi*2G2) where G1 is a complete graph Kn or Cycle Cn, and G2 is K1 or P2. Furthermore, we determined ϒ(G1 *mG2) due to domination number of complete graph Kn and cycle Cn. Since ϒ(Kn) = 1 then ϒ (Kn*mK1) = 1 + m. Furthermore, ϒ(Kn*mP2) = 1 + m. Since ϒ ( C n ) = [ n 3 ] then ϒ ( C n * m K 1 ) = [ n 3 ] + [ n 3 ] m = [ n 3 ] ( 1 + m ) = ϒ ( C n ) ( 1 + m ) . Furthermore, ϒ ( C n * m P 2 ) = [ n 3 ] + [ n 3 ] m = [ n 3 ] ( 1 + m ) = ϒ ( C n ) ( 1 + m ) .
Abstract. Let G = (V, E) be a connected graphs with vertex set V (G), edge set E(G) andwhere d(v, S k ) represents the distance between the vertex v and the set S k , defined byThe minimum resolving partition Π is a partition dimension of G, denoted by pd(G). The resolving partition Π = {S1, S2, S3, . . . , S k } is called a star resolving partition for G if it is a resolving partition and each subgraph induced by Si, 1 ≤ i ≤ k, is a star. The minimum k for which there exists a star resolving partition of V (G)
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.