Conditional resolvability in graphs: a survey

For an ordered set W = { w 1 , w 2 , … , w k } of vertices and a vertex v in a connected graph G , the code of v with respect to W is the k -vector c W ( v ) = ( d ( v , w 1 ) , d ( v , w 2 ) , … , d ( v , w k ) ) , where d ( x , y ) represents the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct codes with respect to W . The minimum cardinality of a resolving set for G is its dimension dim ( G ) . Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs.