On the Polarity and Monopolarity of Graphs

Polarity and monopolarity are properties of graphs defined in terms of the existence of certain vertex partitions; graphs with polarity and monopolarity are respectively called polar and monopolar graphs. These two properties commonly generalize bipartite and split graphs, but are hard to recognize in general. In this article we identify two classes of graphs, triangle‐free graphs and claw‐free graphs, restricting to which provide novel impact on the complexity of the recognition problems. More precisely, we prove that recognizing polarity or monopolarity remains NP‐complete for triangle‐free graphs. We also show that for claw‐free graphs the former is NP‐complete and the latter is polynomial time solvable. This is in sharp contrast to a recent result that both polarity and monopolarity can be recognized in linear time for line graphs. Our proofs for the NP‐completeness are simple reductions. The polynomial time algorithm for recognizing the monopolarity of claw‐free graphs uses a subroutine similar to the well‐known breadth‐first search algorithm and is based on a new structural characterization of monopolar claw‐free graphs, a generalization of one for monopolar line graphs obtained earlier.     

Completing orientations of partially oriented graphs

We initiate a general study of what we call orientation completion problems. For a fixed class of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in by orienting the (nonoriented) edges in P. Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs. We study orientation completion problems for various classes of oriented graphs, including k‐arc‐strong oriented graphs, k‐strong oriented graphs, quasi‐transitive‐oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in‐tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP‐complete. We also show that some of the NP‐complete problems become polynomial time solvable when the input‐oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.     

Scheduling unit-length jobs with precedence constraints of small height

We consider the problem of scheduling unit-length jobs on identical machines subject to precedence constraints. We show that natural scheduling rules fail when the precedence constraints form a collection of stars or a collection of complete bipartite graphs. We prove that the problem is in fact NP-hard on collections of stars when the input is given in a compact encoding, whereas it can be solved in polynomial time with standard adjacency list encoding. On a subclass of collections of stars and on collections of complete bipartite graphs we show that the problem can be solved in polynomial time even when the input is given in compact encoding, in both cases via non-trivial algorithms.     

A Characterization of Mixed Unit Interval Graphs

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half‐open unit intervals of the real line. This is a proper superclass of the well‐known unit interval graphs. Our result solves a problem posed by Dourado, Le, Protti, Rautenbach, and Szwarcfiter (Mixed unit interval graphs, Discrete Math 312, 3357–3363 (2012)).     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

Neighborhood hypergraphs of bipartite graphs

Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP‐hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP‐hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial‐time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

3‐colorability of 4‐regular hamiltonian graphs

On the model of the cycle‐plus‐triangles theorem, we consider the problem of 3‐colorability of those 4‐regular hamiltonian graphs for which the components of the edge‐complement of a given hamiltonian cycle are non‐selfcrossing cycles of constant length ≥ 4. We show that this problem is NP‐complete. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 125–140, 2003     

Completion of the mixed unit interval graphs hierarchy

We describe the missing class of the hierarchy of mixed unit interval graphs. This class is generated by the intersection graphs of families of unit intervals that are allowed to be closed, open, and left‐closed‐right‐open. (By symmetry, considering closed, open, and right‐closed‐left‐open unit intervals generates the same class.) We show that this class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic‐time algorithms that recognize graphs from this class and produce a corresponding interval representation if one exists. We also show that the algorithm from Shuchat et al. [8] directly extends to provide a quadratic‐time algorithm to recognize the class of mixed unit interval graphs.     

The complexity of the matching‐cut problem for planar graphs and other graph classes

The Matching‐Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be ‐complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains ‐complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is ‐complete for planar graphs with girth five. The reduction is from planar graph 3‐colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching‐cuts are described. These classes include claw‐free graphs, co‐graphs, and graphs with fixed bounded tree‐width or clique‐width. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 109–126, 2009     

On Switching to H‐Free Graphs

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H‐free graph. Polynomial‐time algorithms are known for H having at most three vertices or isomorphic to P₄. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP‐complete, thus solving an open problem [Kratochvíl, Ne?et?il and Zýka, Ann Discrete Math 51 (1992)]. Further, we give a characterization of graphs switching equivalent to a K_{1, 2}‐free graph by ten forbidden‐induced subgraphs, each having five vertices. We also give the forbidden‐induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.     

Strongly Regular Decompositions of the Complete Graph

We study several questions about amorphic association schemes and other strongly regular decompositions of the complete graph. We investigate how two commuting edge-disjoint strongly regular graphs interact. We show that any decomposition of the complete graph into three strongly regular graphs must be an amorphic association scheme. Likewise we show that any decomposition of the complete graph into strongly regular graphs of (negative) Latin square type is an amorphic association scheme. We study strongly regular decompositions of the complete graph consisting of four graphs, and find a primitive counterexample to A.V. Ivanov's conjecture which states that any association scheme consisting of strongly regular graphs only must be amorphic.     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].     

Edge‐colorings avoiding rainbow stars

We consider an extremal problem motivated by a article of Balogh [J. Balogh, A remark on the number of edge colorings of graphs, European Journal of Combinatorics 27, 2006, 565–573], who considered edge‐colorings of graphs avoiding fixed subgraphs with a prescribed coloring. More precisely, given , we look for n‐vertex graphs that admit the maximum number of r‐edge‐colorings such that at most colors appear in edges incident with each vertex, that is, r‐edge‐colorings avoiding rainbow‐colored stars with t edges. For large n, we show that, with the exception of the case , the complete graph is always the unique extremal graph. We also consider generalizations of this problem.     

Regular path decompositions of odd regular graphs

Kotzig asked in 1979 what are necessary and sufficient conditions for a d‐regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1‐factor is both necessary and sufficient. Even more, each 1‐factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1‐factor. We conjecture that existence of a 1‐factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1‐factors appear to be extendable and we show that for the family of simple 5‐regular graphs with no cycles of length 4, all 1‐factors are extendable. However, for d>3 we found infinite families of d‐regular simple graphs with non‐extendable 1‐factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5‐regular graphs all 1‐factors are extendable. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 114–128, 2010     

An inverse problem for Sturm‐Liouville operators on trees with partial information given on the potentials

We consider Sturm‐Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class . We suppose that the potentials are known on a part of the graph, and study the so‐called partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Riesz‐basis property of special systems of vector functions and the reduction of the partial inverse problem to the complete one on a part of the graph.     

NP‐completeness of list coloring and precoloring extension on the edges of planar graphs

In the edge precoloring extension problem, we are given a graph with some of the edges having preassigned colors and it has to be decided whether this coloring can be extended to a proper k‐edge‐coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP‐complete on (a) planar 3‐regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series‐parallel graphs. This improves previous results of Easton and Parker 6 , and Fiala 8 . © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 313–324, 2005