The Partition Dimension of Corona Product of Complete and Wheel Graph
Muhammad Ahnaf Yusuf, Hasmawati, Muhammad Rozzaq Hamidi, Andi Muhammad Anwar
TL;DR
Addresses the problem of determining the partition dimension $pd(K_n\circ W_m)$ for corona products of complete graphs and wheel graphs. It employs resolving partitions together with the concepts of equivalent and same-level vertices to derive bounds and construct explicit resolving partitions. The main contributions are exact formulas for $pd(K_n\circ W_m)$ in the cases $m=n$, $m=n+1$, and $m=n+2$, with clear distinctions by small $n$ values. These results advance understanding of partition dimensions for corona-graph families and provide concrete partition schemes useful in graph identification tasks.
Abstract
The graph G is a pair of sets (V(G), E(G)), where V(G) is a finite set whose elements are called vertices, and E(G) is a set of pairs of members of V(G), which is called the edges. Let G be a simple graph. For an ordered k-partition \{Π\} = \{S_1, S_2, \dots, S_k\} of V(G), the representation of u with respect to \{Π\} is k-ordered pairs, r(u \mid \{Π\}) = (d(u, S_1), d(u, S_2), \dots, d(u, S_k)). The partition \{Π\} is called a resolving partition of G if r(u \mid \{Π\}) \neq r(v \mid \{Π\}) for all distinct u, v \in V(G). The resolving partition \{Π\} with the minimum cardinality is called minimum resolving partition. The partition dimension of G, denoted pd(G), is the cardinality of a minimum resolving partition of G. In this research, we determine the partition dimension of the corona product of a complete graph using some mathematical statements about resolving partitions, the concept of equivalent vertices, and same-level vertices. Several analysis results for the K_n \circ W_m vertices refer to equivalent vertices and the same-level vertices concept. The results show that for m = n, pd(K_n \circ W_m) = n, for n \geq 3, for m = n + 1, pd(K_n \circ W_m) = 3 for n = 3, and pd(K_n \circ W_m) = n for n \geq 3. For m = n + 2, pd(K_n \circ W_m) = 4 for n = 2, 3, and pd(K_n \circ W_m) = n for n \geq 4.