On mixed metric dimension in subdivision, middle, and total graphs

Abstract

Let G be a graph and let S(G), M(G), and T (G) be the subdivision, the middle, and the total graph of G, respectively. Let dim(G), edim(G), and mdim(G) be the metric dimension, the edge metric dimension, and the mixed metric dimension  of G, respectively. In this paper, for the subdivision graph it is proved that ½_max{_dim(G), edim(G)} ≤ m_dim(S(G)) ≤  m_dim(G). A family of graphs G_n is constructed for which mdim(G_n) − m_dim(S(G_n)) ≥ 2 holds and this shows that the  inequality mdim(S(G)) ≤ m_dim(G) can be strict, while for a cactus graph G, m_dim(S(G)) = m_dim(G). For the middle graph  it is proved that dim(M(G)) ≤ m_dim(G) holds, and if G is tree with n₁(G) leaves, then _dim(M(G)) = m_dim(G) = n₁(G).  Moreover, for the total graph it is proved that m_dim(T (G)) = 2n₁(G) and _dim(G) ≤ _dim(T (G)) ≤ n₁(G) hold when G is a  tree.