Bounds on the connected local dimension of graphs in terms of the marked dimension and the clique number. Issue 2 (4th May 2022) - Isariyapalakul, Supachoke; Pho-on, Witsarut; Khemmani, Varanoot
Abstract: Let G be a connected graph and let v be a vertex of G . The representation of v with respect to an ordered set W = w 1, w 2, …, w k is the k -vector r ( v | W ) = ( d ( v, w 1 ), d ( v, w 2 ), …, d ( v, w k ) ), where d ( v, w i ) is a distance between v and wi for 1 ≤ i ≤ k . If the representations of any two adjacent vertices of G with respect to W are distinct and the induced subgraph 〈 W 〉 is connected, then W is called a connected local resolving set of G . The minimum cardinality of connected local resolving sets of G is referred to as the connected local dimension of G, denoted by cld ( G ) . A connected local resolving set of cardinality cld ( G ) is called a minimum connected local resolving set or a connected local basis of G . The true twin graph tG of G is obtained by true twin equivalence classes of G such that the vertex set of tG consists of every true twin equivalence class of G and any two distinct vertices of tG are adjacent if the distance of them in G is 1. A connected local resolving set of tG containing all marked vertices is called a marked set of tG . A marked set of tG having minimum cardinality is called a minimum marked set or a marked basis of tG and this cardinality is called the marked dimension of tG, which is denoted by md ( t G ) . In this work, we investigate the connected local dimension of G by using the marked dimension of its true twin graph tG . The bounds for the connected local dimension of G are presented in terms of the marked dimension of tG and the clique number of a set of all marked vertices of tG .