An approach for factoring general boolean functions was described in [15] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring readonce 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 che...

The paper deals with graph operators—the Gallai graphs and the anti-Gallai graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs and the anti-Gallai graphs to be H-free for any finite graph H. The case of complement reducible graphs—cographs is discussed in detail. Some relations between the chromatic number, the radius and the diameter of a graph and it...

Let G = (V,E) be a graph with vertex set V and edge set E and let T be a subset of V . A terminal path cover PC of G with respect to T is a set of pairwise vertex-disjoint paths of G, such that all vertices of G are visited by exactly one path of PC and all vertices in T are end vertices of paths in PC. The terminal path cover problem is to find a terminal path cover of G of minimum cardinality...

Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We first improve Jerrum’s bound using the grundy number, which is the wors...

A sandwich problem for a graph with respect to a graph property Π is a partially specified graph, i.e., only some of the edges and non-edges are given, and the question to be answered is, can this graph be completed to a graph which has the property Π? The graph sandwich problem was investigated for a large number of families of graphs in a 1995 paper by Golumbic, Kaplan and Shamir, and much su...

According to the clan decomposition theorem of Ehrenfeucht and Rozenberg (1990) each labeled 2-structure has a decomposition into three types of basic 2-structures: complete, linear and primitive. This decomposition tan be expressed as a node labeled tree, the shape of the 2-structure. Our main interest is in the uniformly nonprimitive 2-structures, which do not have primitive substructures. Ev...

Spanning tree congestion is a relatively new graph parameter, which has been studied intensively. This paper studies the complexity of the problem to determine the spanning tree congestion for non-sparse graph classes, while it was investigated for some sparse graph classes before. We prove that the problem is NP-hard even for chain graphs and split graphs. To cope with the hardness of the prob...

We give a characterization of DFS cotree-critical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor [2]) by the authors, and deduce a justification of a very simple algorithm for finding a Kuratowski subdivision in a DFS cotree-critical graph. Communicated by: P. Mutzel and M. Jünger; subm...

The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of n nodes. In this paper we consider the situation where the joint input to the nodes is an nnode labeled graph G, i.e., the local input received by each node is the indicator function of its neighborhood in G. Nodes execute an algorithm, communicati...

Classes with bounded rankwidth are MSO-transductions of trees and classes linear paths – a result that shows strong link between the properties these graph considered from point view structural theory finite model theory. We take both views on prove theoretic classes. The results we obtain following. (1) number unlabeled graphs order n rank-width at most r is [(2r+1)(r+1)!2r23r+1]n (2) Graphs l...