Abstract
We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.
Highlights
Interval graphs model the intersection structure of a set of intervals of any linearly ordered structure, and have applications in fields as diverse as VLSI channel routing, molecular biology and scheduling
Our interest in subgraphs of interval graphs stems in part from the completion problem [5] and from the elusiveness of results relating several graph parameters based on linear layouts of graphs
Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11]
Summary
Interval graphs model the intersection structure of a set of intervals of any linearly ordered structure, and have applications in fields as diverse as VLSI channel routing, molecular biology and scheduling. Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11]. ☛ we introduce -proper interval graphs and give some preliminary definitions and results related to these graphs. ☛ ☞ properties of their restricted path-decompositions By varying both and we obtain an infinite lattice of graph classes, ordered by containment. ✪ ☛ Definition 2.1 A graph is a -proper interval graph if it has an interval model such that no interval is ☛ properly contained in more than other intervals. ☛✌ an interval which is 2-sided properly contained in at least other intervals
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