On the star partition dimension of comb product of cycle and complete graph

Let G = (V, E) be a connected graphs with vertex set V(G), edge set E(G) and S (subset of) over bar V (G). For an ordered partition II = {S-1, S-2, S-3, S-k of V (G), the representation of a vertex v is an element of V (G) with respect to II is the k-vectors r(v vertical bar II) = (d(v, S-1), d(v, S-2),..., d(v, S-k)), where d(v, S-k) represents the distance between the vertex v and the set Sk, defined by d(v,S-k) = min{d(v,x) vertical bar x is an element of S-k}. The partition II of V(G) is a resolving partition if the k-vektors r(v vertical bar II), v is an element of V(G) are distinct. The minimum resolving partition II is a partition dimension of G, denoted by pd(G). The resolving partition II = {S-1, S-2, S-3,... S-k is called a star resolving partition for G if it is a resolving partition and each subgraph induced by S-i, 1 <= i <= k, is a star. The minimum k for which there exists a star resolving partition of V(G) is the star partition dimension of G, denoted by spd(G). Finding a star partition dimension of G is classified to be a NP-Hard problem. Furthermore, the comb product between G and H, denoted by G (sic) H, is a graph obtained by taking one copy of G and vertical bar V(G)vertical bar copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. By definition of comb product, we can say that V (G (sic) H) = {(a, u) is an element of V (G),u is an element of V (H)} and (a, u) (b, v) is an element of E(G (sic) H) whenever a = b and uv is an element of E(H), or ab is an element of E(G) and u = v = o. In this paper, we will study the star partition dimension of comb product of cycle and complete graph, namely C-n (sic) K-m and K-m (sic) C-n for n >= 3 and m >= 3.