The Metric Dimension of the Join of Paths and Cycles

Abstract

For an ordered subset $W = \{w_1, w_2, \ldots , w_k \}$ of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector $r(v|W) = (d(v, w_1), \ldots , d(v, w_k ))$, where $d(v,w_i)$ is the distance of the vertices v and $w_i$ in G. The set W is called a resolving set of G if $r(u|W)=r(v|W)$ implies $u=v$. A resolving set of G with minimum cardinality is called a metric basis of G. The metric dimension of G, denoted by $\beta (G)$, is the cardinality of a metric basis of G.

The join of G and H, denoted by $G + H$, is the graph with vertex set $V(G+H) = V(G) \cup V(H)$ and edge set $E(G+H) = E(G) \cup E(H) \cup \{uv : u \in V(G), v \in V(H)\}$. In general, the join of m graphs $G_1, G_2, \ldots , G_m$, denoted by $G = G_1 + G_2 + \cdots + G_m$, has vertex set $V(G) = V(G_1) \cup V(G_2) \cup \cdots \cup V(G_m)$ and edge set $E(G) = E(G_1) \cup E(G_2) \cup \cdots \cup E(G_m) \cup \{uv : u \in G_i, v \in G_j, i \ne j\}$.

In this paper, we compute the metric dimension of the join of a finite number of paths, the join of a finite number of cycles, and the join of paths and cycles.