The metric dimension of k-subdivision graphs

Abstract

The concept of metric dimension of graph could be applied in many graphs, one of them is subdivision graph. A subdivision graph of graph G denoted as S(G) is a graph resulting from graph G by replacing an edge uv with a new vertex w and adding two new edges uw and wv. In this paper, the subdivision graph is called k − subdivision denoted by Sk (G), if the number of edges replaced from graph G is k for 1 ≤ k ≤ |E(G)|, where |E(G)| is the size of graph G. The purpose of this research is to find the metric dimension of subdivision graphs Sk (G), specifically for some special graphs, those are path graph (Pn ), cycle graph (Cn ), complete graph (Kn ), star graph (Sn ), ladder graph (Ln ), book graph (Bn ), wheel graph (Wn ) and fan graph (Fn ).

On the r-dynamic chromatic number of subdivision of wheel graph

“…Based on the research of several researchers above, in this study we will identify r-dynamic chromatic number subdivisions of wheel graphs. Subdivision Graph is defined as a graph derived from graph (H) by deleting the uv edge, replacing the uv edge with the vertex w, and inserting two new edges uw and vv [13]. There are three kinds of subdivision in wheel graphs, namely subdivision on the rim of the wheel graph, subdivision on the gear of the wheel graph, and subdivision on the one cycle of the wheel graph.…”