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 max{dim(G), edim(G)} ≤ mdim(S(G)) ≤ mdim(G). A family of graphs G_n is constructed for which mdim(G_n) − mdim(S(G_n)) ≥ 2 holds and this shows that the inequality mdim(S(G)) ≤ mdim(G) can be strict, while for a cactus graph G, mdim(S(G)) = mdim(G). For the middle graph it is proved that dim(M(G)) ≤ mdim(G) holds, and if G is tree with n₁(G) leaves, then dim(M(G)) = mdim(G) = n₁(G). Moreover, for the total graph it is proved that mdim(T(G)) = 2n₁(G) and dim(G) ≤ dim(T (G)) ≤ n₁(G) hold when G is a tree.

Mathematics Subject Classification (2020):

• 05C12
• 05C69
• 05C76

Key words:

• Resolving set
• mixed resolving set
• edge resolving set
• subdivision graph
• middle graph
• total graph
• tree