## Dissertations

12-1994

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Naveed Sherwani

Dr. Gary Chartrand

Dr. Yousef Alavi

Dr. Dionysios Kountanis

#### Abstract

Physical design is one of several stages in the design of a VLSI chip. In this stage, the specifications of an electrical circuit are converted into a geometrical model. Problems concerning the physical design stage can often be studied by means of graphs. The problems encountered here are routing problems and concern placement of vertices, which represent wires, into layers. All this gives rise to a class of graphs whose vertices are partitioned into classes. Such graphs are called stratified graphs. In this dissertation, we formally define stratified graphs, study their properties, and investigate various algorithmic problems related to these graphs.

In Chapter I, the concept of stratified graphs is formally defined and basic terminology and notation are introduced. Chapter II concerns degrees and degree multisets in stratified graphs. The concept of color pattern is defined. Each vertex of a stratified graph has, in fact, a unique color pattern. Those multisets of color patterns, called arrangements, that are realized in a stratified graph are characterized. Color-regular graphs are defined and, with small restriction, it is shown that color-regular 2-stratified graphs exist with prescribed color patterns, and, in fact, the minimum order of such graphs is determined. It is shown that the so-called stratification problem that arises in this context is NP-complete.

In Chapter III, the emphasis is on distance in stratified graphs. The concepts of eccentricity, radius, diameter, center, and periphery are defined for these graphs and investigated, as well as other distance-related concepts. It is shown that for every set S = {H1, H2 , ..., Hk} of K:-stratified graphs, there exists a k stratified graph G whose k centers axe predsely the graphs in S. Those k-stratified graphs that are peripheries of k-stratified graphs are characterized. The distancerelated concepts of proximity and seclusion, for which there is no analogue in ordinary graphs, is introduced and studied.

Chapter IV deals with a variety of algorithmic problems related to stratified graphs. A dynamic programming algorithm is presented for determining a smallest color-spanning subtree. In Chapter V, specific routing problems from the VLSI physical design stage are modeled by stratified graphs and related algorithms are presented. The dissertation concludes in Chapter VI with a discussion of open problems and some directions for future research.