International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 38, Pages 1997-2017
doi:10.1155/S0161171204311403
Conditional resolvability in graphs: a survey
1Department of Mathematics, Srinakharinwirot University, Bangkok 10110, Thailand
2Department of Mathematics, Western Michigan University, Kalamazoo 49008, MI, USA
Received 29 November 2003
Copyright © 2004 Varaporn Saenpholphat and Ping Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For an ordered set W={w1,w2,…,wk} of vertices and
a vertex v in a connected graph G, the code of v with
respect to W is the k-vector cW(v)=(d(v,w1),d(v,w2),…,d(v,wk)), 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.