Distance in stratified graphs

Abstract

A graph G is stratified if its vertex set is partitioned into classes, called strata. If there are k strata, then G is k-stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color X in a stratified graph G, the X-eccentricity e X(v) of a vertex v of G is the distance between v and an X-colored vertex furthest from v. The minimum X-eccentricity among the vertices of G is the X-radius radX G of G and the maximum X-eccentricity is the X-diameter diamX G. It is shown that for every three positive integers a, b and k with a≤b, there exist a k-stratified graph G with radX G = a and diamX G = b. The number s X denotes the minimum X-eccetricity among the X-colored vertices of G. It is shown that for every integer t with radX G ≤ t ≤ diamX G, there exist at least one vertex v with e X(v) = t; while if radX G ≤ t ≤ s X, then there are at least two such vertices. The X-center C X(G) is the subgraph induced by those vertices v with e X(v) = radX G and the X-periphery P X (G) is the subgraph induced by those vertices v with e X(G) = diamX G. It is shown that for k-stratified graphs H 1, H 2,..., H k with colors X 1, X 2,..., X k and a positive integer n, there exists a k-stratified graph G such that C X i(G)≅ H i (1 ≤; i ≤; k1) and \(d(C_{x_i } (G),C_{x_j } (G)) \geqslant n{\text{ for }}i \ne j.\) for i ≠ j. Those k-stratified graphs that are peripheries of k-stratified graphs are characterized. Other distance-related topics in stratified graphs are also discussed.