Combinatorics,graph theory,and computing

Journal of Algebraic Combinatorics -     

An improvement on the complexity of factoring read-once Boolean functions

Read-once functions have gained recent, renewed interest in the fields of theory and algorithms of Boolean functions, computational learning theory and logic design and verification. In an earlier paper [M.C. Golumbic, A. Mintz, U. Rotics, Factoring and recognition of read-once functions using cographs and normality, and the readability of functions associated with partial k-trees, Discrete Appl. Math. 154 (2006) 1465-1677], we presented the first polynomial-time algorithm for recognizing and factoring read-once functions, based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrence graph is P₄-free.In this note, we improve the complexity bound by showing that the method can be modified slightly, with two crucial observations, to obtain an O(n|f|) implementation, where |f| denotes the length of the DNF expression of a positive Boolean function f, and n is the number of variables in f. The previously stated bound was O(n²k), where k is the number of prime implicants of the function. In both cases, f is assumed to be given as a DNF formula consisting entirely of the prime implicants of the function.     

Comparability graphs and a new matroid

We consider a variety of two person perfect information games of the following sort. On the ith round Player I selects a vector v_i of a certain prescribed form and Player II either adds or subtracts v_i from a cumulative sum. Player II's object is to keep the cumulative sum as small as possible. We give bounds on the value of such games under a variety of conditions.     

Tolerance graphs

Tolerance graphs arise from the intersection of intervals with varying tolerances in a way that generalizes both interval graphs and permutation graphs. In this paper we prove that every tolerance graph is perfect by demonstrating that its complement is perfectly orderable. We show that a tolerance graph cannot contain a chordless cycle of length greater than or equal to 5 nor the complement of one. We also discuss the subclasses of bounded tolerance graphs, proper tolerance graphs, and unit tolerance graphs and present several possible applications and open questions.     

Algorithmic aspects of intersection graphs and representation hypergraphs

Let be a family of sets. The intersection graph of is obtained by representing each set in by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The grandfather of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.The scope of research that has been going on in this general area extends from the mathematical and algorithmic properties of intersection graphs, to their generalizations and graph parameters motivated by them. In addition, many real-world applications involve the solution of problems on such graphs.In this paper a number of topics in algorithmic combinatorics which involve intersection graphs and their representative families of sets are presented. Recent applications to computer science are also discussed. The intention of this presentation is to provide an understanding of the main research directions which have been investigated and to suggest possible new directions of research.     

A characterization of chain probe graphs

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+F is a chain graph (i.e., a 2K ₂-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n ²) recognition algorithm.     

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that have an (h,s,t)-representation is denoted by [h,s,t]. It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below.In this paper, we investigate the class of [h,2,t] graphs, i.e., the intersection graphs of paths in a tree. The [h,2,1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h,2,2] graphs are known as the EPT graphs. We consider variations of [h,2,t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h,2,t] and orthodox-[h,2,t] graphs for varied values of h and t.     

The edge intersection graphs of paths in a tree

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.     

Path-Bicolorable Graphs

In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V, E) is P _k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., color classes) V ₁ and V ₂ such that for every induced P _k (a path with exactly k − 1 edges and k vertices) in G, the two colors alternate along the P _k , i.e., no two consecutive vertices of the P _k belong to the same color class V _i , i = 1, 2. Obviously, a graph is bipartite if and only if it is P ₂-bicolorable. We give a structural characterization of P ₃-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P ₄-bicolorable graphs in terms of forbidden subgraphs.