Damon Kaller

Definability Equals Recognizability of Partial 3-Trees and \sl k -Connected Partial \sl k -Trees

Abstract. We consider graph decision problems on partial 3-trees that can be solved by a finite-state, leaf-to-root tree automaton processing a width-3 tree decomposition of the given graph. The class of yes-instances of such a problem is said to be recognizable by the tree automaton. In this paper we show that any such class of recognizable partial 3-trees is definable by a sentence in CMS logic. Here, CMS logic is the extension of Monadic Second-order logic with predicates to count the cardinality of sets modulo fixed integers. We also generalize this result to show that recognizability (by a tree automaton that processes width-k tree decompositions) implies CMS-definability for k -connected partial k -trees. The converse—definability implies recognizability—is known to hold for any class of partial k -trees, and even for any graph class with an appropriate definition of recognizability. It has been conjectured that recognizability implies definability over partial k -trees; but a proof was previously known only for k ≤ 2 . This paper proves the conjecture, and hence the equivalence of definability and recognizability, over partial 3-trees and k -connected partial k -trees. We use techniques that may lead to a proof of this equivalence over all partial k -trees.

AnO(m+nlogn) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m+nlogn) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(mloglogn+nlogn), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the loglognfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(mloglogn+nlogn) algorithm for finding a maximum-weight clique?which matches the best known algorithm.

Definability Equals Recognizability of Partial 3-Trees

We show that a graph decision problem can be defined in the Counting Monadic Second-order logic if the partial 3-trees that are yes-instances can be recognized by a finite-state tree automaton. The proof generalizes to also give this result for k-connected partial k-trees. The converse—definability implies recognizability—is known to hold over all partial k-trees. It has been conjectured that recognizability implies definability over partial k-trees; but a proof was previously known only for k≤2. This paper proves the conjecture—and hence the equivalence of definability and recognizability—over partial 3-trees and k-connected partial k-trees.

On the Complements of Partial k-Trees

We introduce new techniques for studying the structure of partial k-trees. In particular, we show that the complements of partial k-trees provide an intuitively-appealing characterization of partial k-tree obstructions. We use this characterization to obtain a lower bound of 2Ω(k log k) on the number of obstructions, significantly improving the previously best-known bound of 2Ω(√k). Our techniques have the added advantage of being considerably simpler than those of previous authors.

Vertex Partitioning Problems On Partial k-Trees

We describe a general approach to obtain polynomial-time algorithms over partial k-trees for graph problems in which the vertex set is to be partitioned in some way. We encode these problems with formulae of the Extended Monadic Second-order (or EMS) logic. Such a formula can be translated into a polynomial-time algorithm automatically. We focus on the problem of partitioning a partial k-tree into induced subgraphs isomorphic to a fixed pattern graph; a distinct algorithm is derived for each pattern graph and each value of k. We use a “pumping lemma” to show that (for some pattern graphs) this problem cannot be encoded in the “ordinary” Monadic Second-order logic—from which a linear-time algorithm over partial k-trees would be obtained. Hence, an EMS formula is in some sense the strongest possible. As a further application of our general approach, we derive a polynomial-time algorithm to determine the maximum number of co-dominating sets into which the vertices of a partial k-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).

Regular-Factors In The Complements Of Partial k-Trees

We consider the problem of recognizing graphs containing an f-factor (for any constant f) over the class of partial k-tree complements. We also consider a variation of this problem that only recognizes graphs containing a connected f-factor: this variation generalizes the Hamiltonian circuit problem. We show that these problems have O(n) algorithms for partial k-tree complements (on n vertices); we assume that the Θ(n2) edges of such a graph are specified by representing the O(n) edges of its complement. As a preliminary result of independent interest, we demonstrate a logical language in which, if a graph property can be expressed over the class of partial k-tree complements, then those graphs that satisfy the property can be recognized in O(n) time.

The χt-coloring problem

Motivated by a problem in Scheduling Theory, we introduce the χt-coloring problem, a generalization of the chromatic number problem that places a bound of t on the size of any color class. For fixed t, we show that the perfect χt-coloring problem (in which each color class must have cardinality exactly t) can be expressed in the counting monadic second-order logic and, hence, has a linear-time algorithm over the class of graphs G of bounded treewidth: A solution is a partition of G into induced subgraphs, each isomorphic to a fixed graph consisting of t isolated vertices. The logical formalism generalizes to allow these t vertices to be t isomorphic connected components. The linear-time algorithm so derived for the perfect χt-coloring problem is used to design a linear-time algorithm for the optimization version of the general χt-coloring problem (for fixed t) on graphs of bounded treewidth. We also show that this problem has a polynomial-time algorithm on bipartite graphs.