On the partition dimension of comb product of path and complete graph.

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 II = {S1; S2; S3; ..., Sk} of the vertex set V of G, the representation of a vertex v 2 V with respect to II is the vector r(vII) = (d(v; S1); d(v; S2); ..., d(v; Sk)), where d(v; Sk) represents the distance between the vertex v and the set Sk and d(v; Sk) = minfd(v; x)jx 2 Skg. A partition of V(G) is a resolving partition if different vertices of G have distinct representations, i.e., for every pair of vertices ¹ v 2 V(G); r(ujΓ), r(vjΓ). The minimum k of 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 Km and path Pn namely pd(Km > Pn) = m where m ≥ 3 and n ≥ 2 and pd(Pn > Km) = m where m ≥ 3, n ≥ 2 and m ≥ n. [ABSTRACT FROM AUTHOR]