On the equivalence in complexity among basic problems on bipartite and parity graphs

Metric characterization of parity graphs

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

On the contour of graphs

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Dominating induced matching in some subclasses of bipartite graphs

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

Even though Kemeny’s constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny’s constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny’s constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time.

Bipartite bithreshold graphs

For a connected graph $$G = \left( V,E\right) $$ , a set $$S\subseteq E(G)$$ is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number $$m_{tev}(G)$$ of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size $$q \ge 3 $$ with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers $$r_{m},d_{m}$$ and $$l\ge 4$$ with $$r_{m}< d_{m} \le 2 r_{m}$$ , there exists a connected graph G with $$\textit{rad}_ {m} G = r_{m}$$ , $$\textit{diam}_ {m} G = d_{m}$$ and $$m_{tev}(G) = l$$ and also shown that for every integers a and b with $$2 \le a \le b$$ , there exists a connected graph G such that $$ m_{ev}\left( G\right) = b$$ and $$m_{tev}(G) = a + b$$ . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by $$f_{tev}(S)$$ is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by $$f_{tev}(G) = \textit{min}\{f_{tev}(S)\}$$ , where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with $$0 \le a \le b$$ and $$b \ge 2$$ , there exists a connected graph G such that $$f_{tev}(G) = a$$ and $$ m _{tev}(G) = b$$ , where $$ f _{tev}(G)$$ is the forcing total edge-to-vertex monophonic number of G.

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

Classes of bipartite graphs related to chordal graphs

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.