Varaporn Saenpholphat

CONDITIONAL RESOLVABILITY IN GRAPHS: A SURVEY

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.

CONNECTED PARTITION DIMENSIONS OF GRAPHS

For a vertex v of a connected graph G and a subset S of V (G), the distance between v and S is d(v;S) = minfd(v;x)jx 2 Sg. For an ordered k-partition ƒ = fS1;S2;¢¢¢;Skg of V (G), the representation of v with respect to ƒ is the k-vector r(vjƒ) = (d(v;S1), d(v;S2);¢¢¢; d(v;Sk)). The k-partition ƒ is a resolving partition if the k-vectors r(vjƒ), v 2 V (G), are distinct. The minimum k for which there is a resolving k-partition of V (G) is the partition dimension pd(G) of G. A resolving partition ƒ = fS1;S2;¢¢¢;Skg of V (G) is connected

ON RESOLVING EDGE COLORINGS IN GRAPHS

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph.