Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [C.H.C. Little, A characterization of convertible (0,1)-matrices, J. Combin. Theory Ser. B 18 (1975) 187–208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K_3,3. We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion.We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.     

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti‐vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d‐factors in bipartite graphs, and perfect 2‐matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Partition the vertices of a graph into one independent set and one acyclic set

For a given graph G, if the vertices of G can be partitioned into an independent set and an acyclic set, then we call G a near-bipartite graph. This paper studies the recognition of near-bipartite graphs. We give simple characterizations for those near-bipartite graphs having maximum degree at most 3 and those having diameter 2. We also show that the recognition of near-bipartite graphs is NP-complete even for graphs where the maximum degree is 4 or where the diameter is 4.     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.     

Maximum weight edge-constrained matchings

Practical questions arising from (for instance) biological applications can often be expressed as classical optimization problems with specific, new features. We are interested here in the version of the maximum weight matching problem (on a graph G) obtained by (1) defining a set F of pairs of incompatible edges of G and (2) asking that the matching contains at most one edge in each given pair. Such a matching is called an odd matching. The graph T(F)=(V_F,F), where V_F is the set of edges of G occurring in at least one pair of F, is called the trace-graph of G and F.We motivate the introduction of the maximum weight odd-matching (abbreviated as Odd-MWM) problem and study its complexity with respect to two parameters: the type of graph G and the graph class T to which T(F) belongs.Our contribution includes:

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A proof that Odd-MWM is NP-complete for 3-degree bipartite graphs when T(F) is a matching (i.e. when T is the class of 1-regular graphs), even if the weight function is constant.

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A proof that Odd-MWM is NP-complete (for 3-degree bipartite graphs as well as for any larger class) if and only if T is a class of graphs with unbounded induced matching. Otherwise, Odd-MWM is polynomial.

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A (Δ(T(F))+1)-approximate algorithm for Odd-MWM on general graphs. This algorithm becomes a χ(T(F))-approximate algorithm when the graph class T admits a polynomial algorithm for minimum vertex coloring.

     

Extremal perfect graphs for a bound on the domination number

We examine classes of extremal graphs for the inequality γ(G)?|V|-max{d(v)+β_v(G)}, where γ(G) is the domination number of graph G, d(v) is the degree of vertex v, and β_v(G) is the size of a largest matching in the subgraph of G induced by the non-neighbours of v. This inequality improves on the classical upper bound |V|-maxd(v) due to Claude Berge. We give a characterization of the bipartite graphs and of the chordal graphs that achieve equality in the inequality. The characterization implies that the extremal bipartite graphs can be recognized in polynomial time, while the corresponding problem remains NP-complete for the extremal chordal graphs.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

Jump number of dags having Dilworth number 2

The jump number, denoted by σ, of a directed acyclic graph (dag) G, is the minimum number of arcs that have to be added to G such that the resulting graph is still acyclic and has a hamiltonian path.We study here the particular class of dags having an induced partial order of width 2, and give a characterization of such graphs with σ(G)=i. This yields immediately a polynomial algorithm to compute the jump number in this particular class.     

A characterization of triangle-free tolerance graphs

We prove that a triangle-free graph G is a tolerance graph if and only if there exists a set of consecutively ordered stars that partition the edges of G. Since tolerance graphs are weakly chordal, a tolerance graph is bipartite if and only if it is triangle-free. We, therefore, characterize those tolerance graphs that are also bipartite. We use this result to show that in general, the class of interval bigraphs properly contains tolerance graphs that are triangle-free (and hence bipartite).     

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

Matching signatures and Pfaffian graphs

Perfect matchings of k-Pfaffian graphs may be enumerated in polynomial time on the number of vertices, for fixed k. In general, this enumeration problem is #P-complete. We give a Composition Theorem of 2r-Pfaffian graphs from r Pfaffian spanning subgraphs. Constructions of k-Pfaffian graphs known prior to this seem to be of a very different and essentially topological nature. We apply our Composition Theorem to produce a bipartite graph on 10 vertices that is 6-Pfaffian but not 4-Pfaffian. This is a counter-example to a conjecture of Norine (2009) [8], which states that the Pfaffian number of a graph is a power of four.     

Optimal graphs for chromatic polynomials

Let F(n,e) be the collection of all simple graphs with n vertices and e edges, and for G∈F(n,e) let P(G;λ) be the chromatic polynomial of G. A graph G∈F(n,e) is said to be optimal if another graph H∈F(n,e) does not exist with P(H;λ)?P(G;λ) for all λ, with strict inequality holding for some λ. In this paper we derive necessary conditions for bipartite graphs to be optimal, and show that, contrarily to the case of lower bounds, one can find values of n and e for which optimal graphs are not unique. We also derive necessary conditions for bipartite graphs to have the greatest number of cycles of length 4.     

Weakly bipartite graphs and the Max-cut problem

A new class of graphs, called weakly bipartite graphs, is introduced. A graph is called weakly bipartite if its bipartite subgraph polytope coincides with a certain polyhedron related to odd cycle constraints. The class of weakly bipartite graphs contains for instance the class of bipartite graphs and the class of planar graphs. It is shown that the max-cut problem can be solved in polynomial time for weakly bipartite graphs. The polynomical algorithm presented is based on the ellipsoid method and an algorithm that computes a shortest path of even length.