Multiset dimension of kayak paddles graph and cycles with chord

Abstract

Suppose the set $W=\{s_1, s_2,\dots, s_k \}$ is a subset of the vertex set $V(G)$. The representation of a vertex $v$ of $G$ with respect to $W$ as follows \[r_m(v|W)=\{d(v,s_1), d(v,s_2),\dots, d(v,s_k)\}\] where $d(v,s_i)$ is the distance between the vertex $v$ with the vertices of set $W$ together with their multiplicities. The set $W$ is called the {\it m-resolving set} of $G$ if every vertices of $G$ have distinct representation with respect to $W$. If $G$ has an m-resolving set, then an m-resolving set having minimum cardinality is called a multiset basis and its cardinality is called the multiset dimension of $G$, denoted by $md(G)$. We say that $G$ has an infinite multiset dimension and we write $md(G)=\infty$. In this paper, we determine the multiset dimension of kayak paddles graph and cycles with chord.