Partial K-trees Research Articles

Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195–198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131–153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality. For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2 k function and a read 2 k formula for F can be obtained in polynomial time.

Vertex rankings of chordal graphs and weighted trees

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.

Read-Once Functions Revisited and the Readability Number of a Boolean Function

Abstract In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its correctness is 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-occurrance graph is P4-free. We also investigate the problem of factoring certain non-read-once functions. In particular, we show that if the co-occurrence graph of a positive Boolean function f is a tree, then the function is read-twice. We then extend this further proving that if f is normal and its corresponding graph is a partial k-tree, then f is a read 2k function and a read 2k formula for F for f can be obtained in polynomial time.

Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an “almost uniform” partition is called an ( l , u ) -partition. We deal with three problems to find an ( l , u ) -partition of a given graph; the minimum partition problem is to find an ( l , u ) -partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an ( l , u ) -partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O ( u 4 n ) and the p-partition problem can be solved in time O ( p 2 u 4 n ) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.

Embeddings of [formula omitted]-connected graphs of pathwidth [formula omitted

We present O ( n 3 ) embedding algorithms (subgraph isomorphism and its generalizations) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomial-time algorithm for minor containment and the first O ( n c ) algorithm ( c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees, as well as an O ( n 2 ) algorithm for subgraph isomorphism. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions. Finally, we describe n k + O ( 1 ) algorithms for solving these problems on arbitrary graphs of pathwidth at most k.

Finding the maximum common subgraph of a partial k-tree and a graph with a polynomially bounded number of spanning trees

The maximum common subgraph problem is known to be NP-hard, although it has often been applied to various areas. In the field of molecular biology, we can reduce the problem space by analyzing the structures of chemical compounds. In doing so, we have found that the tree-width of chemical compounds are bounded by a constant, and that the possible spanning trees of any compound is polynomially bounded. We present a polynomial time algorithm for finding the maximum common connected induced subgraph of a degree-bounded partial k-tree and a connected graph, the number of whose possible spanning trees is polynomial.

Minimal reducible bounds for induced-hereditary properties

Let ( M a ,⊆) and ( L a ,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈ M a ( ∈ L a ) is called a minimal reducible bound for a property P∈ M a ( ∈ L a ) if in the interval ( P, R) of the lattice M a ( L a ) there are only irreducible properties. The set of all minimal reducible bounds of a property P∈ M a in the lattice M a we denote by B M( P) . Analogously, the set of all minimal reducible bounds of a property P∈ L a in L a is denoted by B L( P) . We establish a method to determine minimal reducible bounds for additive degenerate induced-hereditary (hereditary) properties of graphs. We show that this method can be successfully used to determine already known minimal reducible bounds for k-degenerate graphs and outerplanar graphs in the lattice L a . Moreover, in terms of this method we describe the sets of minimal reducible bounds for partial k-trees and the graphs with restricted order of components in L a and k-degenerate graphs in M a .

A polynomial time algorithm for strong edge coloring of partial k-trees

A matching M in a graph is called induced if there is no edge in the graph connecting two edges of M. The strong edge coloring problem is to find an edge coloring of a given graph with minimum number of colors such that each color class is an induced matching. This problem is known to be NP-complete, even in very restricted cases. Here, we show that it can be solved in polynomial time on graphs with bounded treewidth, i.e partial k-trees. This answers an open question of Mahdian (Discrete Appl. Math. 118 (2002) 239).

On synthesis of easily testable (k, K) circuits

A (k, K) circuit is one which can be decomposed into nonintersecting blocks of gates where each block has no more than K external inputs, such that the graph formed by letting each block be a node and inserting edges between blocks if they share a signal line, is a partial k-tree. (k, K) circuits are special in that they have been shown to be testable in time polynomial in the number of gates in the circuit, and are useful if the constants k and K are small. We demonstrate a procedure to synthesise (k, K) circuits from a special class of Boolean expressions.

A simple competitive graph coloring algorithm II

We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set of colors X, with |X|=r. A color α∈X is a legal color for uncolored vertex v if by coloring v with color α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is a partial k-tree, r=k+1, and d⩾4k−1, then Alice has a winning strategy for this game. In the special case that k=1, this answers a question of Chou, Wang, and Zhu. We also analyze this strategy for other classes of graphs. In particular, we show that there exists a positive integer d such that Alice can win the game on any planar graph if r=6.

Marking Games and the Oriented Game Chromatic Number of Partial k -Trees

Nesetřil and Sopena introduced the concept of oriented game chromatic number. They asked whether the oriented game chromatic number of partial k-trees was bounded. Here we answer their question positively.

Partial k-trees with maximum chromatic number

The paper is concerned with partial k-trees whose chromatic number is maximal, i.e. equal to ( k+1). We have proved that any such graph contains a triangle (if k⩾3), but need not contain a clique on ⌊ k+5 2 ⌋ vertices as a subgraph.

Counting H-colorings of partial k-trees

The problem of counting all H-colorings of a graph G with n vertices is considered. While the problem is, in general, #P-complete, we give linear time algorithms that solve the main variants of this problem when the input graph G is a k-tree or, in the case where G is directed, when the underlying graph of G is a k-tree. Our algorithms remain polynomial even in the case where k=O(logn) or in the case where the size of H is O(n). Our results are easy to implement and imply the existence of polynomial time algorithms for a series of problems on partial k-trees such as core checking and chromatic polynomial computation.

Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees

We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sum-of-completion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial k-trees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NP-hard on planar graphs, even in the unweighted case, known as the sum coloring problem. For a nonpreemptive multicoloring sum of partial k-trees, we obtain a fully polynomial time approximation scheme. This is based on a pseudo-polynomial time algorithm that holds for a general class of cost functions. Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial k-trees that uses only O(logn) preemptions. These results are based on several properties of multicolorings and tools for manipulating them, which may be of more general applicability.

The edge-disjoint paths problem is NP-complete for series–parallel graphs

Many combinatorial problems are NP-complete for general graphs. However, when restricted to series–parallel graphs or partial k-trees, many of these problems can be solved in polynomial time, mostly in linear time. On the other hand, very few problems are known to be NP-complete for series–parallel graphs or partial k-trees. These include the subgraph isomorphism problem and the bandwidth problem. However, these problems are NP-complete even for trees. In this paper, we show that the edge-disjoint paths problem is NP-complete for series–parallel graphs and for partial 2-trees although the problem is trivial for trees and can be solved for outerplanar graphs in polynomial time.

Algorithms for generalized vertex-rankings of partial [formula omitted]-trees

A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. A c-vertex-ranking is optimal if the number of labels used is as small as possible. We present sequential and parallel algorithms to find an optimal c-vertex-ranking of a partial k-tree, that is, a graph of treewidth bounded by a fixed integer k. The sequential algorithm takes polynomial-time for any positive integer c. The parallel algorithm takes O( log n) parallel time using a polynomial number of processors on the common CRCW PRAM, where n is the number of vertices in G.

Maximum packing for [formula omitted]-connected partial [formula omitted]-trees in polynomial time

The problem of determining the maximum number of node-disjoint subgraphs of a partial k-tree H on n H nodes that are isomorphic to a k-connected partial k-tree G on n G nodes is shown to be solvable in time O(n G k+1n H+n H k) .

The game coloring number of pseudo partial k-trees

This paper introduces a new class of graphs: ( a, b)-pseudo partial k-trees. In some sense, the parameters ( a, b) measure a graph's deviation from being a partial k-tree. In particular, a (0,0)-pseudo partial k-tree is just a partial k-tree. We discuss the game coloring number (as well as the game chromatic number) of ( a, b)-pseudo partial k-trees, and prove that the game coloring number of an ( a, b)-pseudo partial k-tree is at most 3 k+2 a+ b+2. In particular, the game coloring number of a partial k-tree is at most 3 k+2. This reduces considerably the previous known upper bound for the game chromatic number of a partial k-tree. Then we describe another strategy for Alice for playing the game, which gives a better upper bound for the game coloring number of ( a, b)-pseudo partial 2-trees. Namely, we prove that the game coloring number of an ( a, b)-pseudo partial 2-tree is at most a+ b+8. By using this result, we prove that the game coloring number of a graph embeddable on an orientable surface of genus g⩾1 is at most ⌊ 1 2 (3 1+48g +23)⌋. This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chromatic number of such graphs. Moreover, the strategy for Alice introduced in this paper for playing the game can be considered as a common generalization of the previous known strategies for playing the game on forests, interval graphs, outerplanar graphs and planar graphs.

A POLYNOMIAL-TIME ALGORITHM FOR FINDING TOTAL COLORINGS OF PARTIAL k-TREES

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees, that is, graphs of treewidth bounded by a constant k. However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a given partial k-tree with the minimum number of colors. This paper gives such a first polynomial-time algorithm.

Edge-Coloring and f-Coloring for Various Classes of Graphs

In an ordinary edge-coloring of a graph G=(V, E) each color appears at each vertex v e V at most once. An f-coloring is a generalized coloring in which each color appears at each vertex v e V at most f(v) times. This paper gives efficient sequential and parallel algorithms which find ordinary edge-colorings and f-colorings for various classes of graphs such as bipartite graphs, planar graphs, graphs of fixed genus, partial k-trees, s-degenerate graphs, graphs of fixed arboricity etc.