WEAK TOTAL RESOLVABILITY IN GRAPHS

Identifier:  imarina:9295799

Handle: https://hdl.handle.net/20.500.11797/imarina9295799

Authors:  Casel, Katrin; Estrada-Moreno, Alejandro; Fernau, Henning; Alberto Rodriguez-Velazquez, Juan

Abstract:
A vertex v is an element of V(G) is said to distinguish two vertices x, y is an element of V(G) of a graph G if the distance from v to x is different from the distance from v to y. A set W subset of V(G) is a total resolving set for a graph G if for every pair of vertices x, y is an element of V(G), there exists some vertex w is an element of W {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x is an element of V(G) W and y is an element of W, there exists some w is an element of W {y} which distinguishes x and y. A weak total resolving set of minimum cardinality is called a weak total metric basis of G and its cardinality the weak total metric dimension of G. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called weak total adjacency dimension and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.