Recognition of some perfectly orderable graph classes

AbstractThis paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P4-simplicial graphs are recognized in O(n3.376) time, improving the previous bounds of O(n4) and O(n5), respectively. Brittle and semi-simplicial graphs are recognized in O(n3) time using a randomized algorithm, and O(n3log2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O(m2). Welsh–Powell opposition graphs are recognized in O(n3) time, improving the previous bound of O(n4). HHP-free graphs and maxibrittle graphs are recognized in O(mn) and O(n3.376) time, respectively

Isomorphism of graph classes related to the circular-ones property

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure

Efficient graph representations

The book deals with questions which arise from storing a graph in a computer. Different classes of graphs admit different forms of computer representations, and focusing on the representations gives a new perspective on a number of problems. For a variety of classes of graphs, the book considers such questions as existence of good representations, algorithms for finding representations, questions of characterizations in terms of representation, and how the representation affects the complexity of optimization problems. General models of efficient computer representations are also considered.

Modular decomposition and transitive orientation

A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n + m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problem