Martin Charles Golumbic

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every

Complexity and algorithms for reasoning about time: a graph-theoretic approach

n\in {\mathcal N}

Graph sandwich problems

there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid,

Four Strikes Against Physical Mapping of DNA

n\in {\mathcal N}

Trapezoid graphs and their coloring

, n ≥ 3, has clique–width exactly n+1.

The edge intersection graphs of paths in a tree

Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions, and all realizations problems are considered for temporal (internal) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NP-complete

Trivially perfect graphs

Abstract The graph sandwich problem for property Π is defined as follows: Given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) such that E 1 ⊆ E 2 , is there a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 which satisfies property Π? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NP-completeness of others. We describe polynomial time algorithms for threshold graphs, split graphs, and cographs. For the sandwich problem for threshold graphs, the only case in which a previous algorithm existed, we obtain a faster algorithm. NP-completeness proofs are given for comparability graphs, permutation graphs, and several other families. For Eulerian graphs; one Version of the problem is polynomial and another is NP-complete.

New results on induced matchings

Physical mapping is a central problem in molecular biology and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NP-complete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the k-consecutive ones problem for k > or = 2. These models have been chosen to reflect various features typical in biological data, including false-negative and positive errors, small width of the map, and chimericism.

Stability in circular arc graphs

Abstract We define trapezoid graphs, an extension of both interval and permutation graphs. We show that this new class properly contains the union of the two former classes, and that trapezoid graphs are equivalent to the incomparability graphs of partially ordered sets having interval order dimension at most two. We provide an optimal coloring algorithm for trapezoid graphs that runs in time O(nk), where n is the number of nodes and k is the chromatic number of the graph. Our coloring algorithm has direct applications to channel routing on integrated circuits.

The complexity of comparability graph recognition and coloring

Abstract The class of edge intersection graphs of a collection of paths in a tree (EPT graphs) is investigated, where two paths edge intersect if they share an edge. The cliques of an EPT graph are characterized and shown to have strong Helly number 4. From this it is demonstrated that the problem of finding a maximum clique of an EPT graph can be solved in polynomial time. It is shown that the strong perfect graph conjecture holds for EPT graphs. Further complexity results follow from the observation that every line graph is an EPT graph. The class of EPT graphs is equivalent to the class of fundamental cycle graphs.