On weighted efficient total domination

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

We study the problem of determining whether a given graph \(G=(V,E)\) admits a matching M whose removal destroys all odd cycles of G (or equivalently whether \(G-M\) is bipartite). This problem is equivalent to determine whether G admits a (2, 1)-coloring, which is a 2-coloring of V(G) in which each color class induces a graph of maximum degree at most 1. We determine a dichotomy related to the NP-completeness of such a decision problem, where it is NP-complete even for 3-colorable planar graphs of maximum degree 4, while it is linear-time solvable for graphs of maximum degree at most 3. In addition, we present polynomial-time algorithms for many graph classes including those in which every odd-cycle subgraph is a triangle, graphs having bounded dominating sets, and \(P_5\)-free graphs. Additionally, we show that this problem is fixed-parameter tractable when parameterized by the clique-width, which implies that it is polynomial-time solvable for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs.

Mind the independence gap

For an integer \(d\ge 1\), let d-claw stand for \(K_{1,d}\), the complete bipartite graph with 1 and d vertices on each part. The d-claw vertex deletion problem, \({d}\text {-}\textsc {claw}\text {-}\textsc {vd}\), asks for a given graph G and an integer k whether one can delete at most k vertices from G such that the resulting graph has no d-claw as an induced subgraph. Thus, \({1}\text {-}\textsc {claw}\text {-}\textsc {vd}\) and \({2}\text {-}\textsc {claw}\text {-}\textsc {vd}\) are just the famous vertex cover problem and the cluster-vd problem, respectively.In this paper, we show that cluster-vd remains \(\mathsf {NP}\)-complete when restricted to bipartite graphs of maximum degree 3. Moreover, for every \(d\ge 3\), we show that \({d}\text {-}\textsc {claw}\text {-}\textsc {vd}\) is \(\mathsf {NP}\)-complete even when restricted to bipartite graphs of maximum degree d. These hardness results are optimal with respect to degree constraint. We also show that, for every \(d\ge 3\), \({d}\text {-}\textsc {claw}\text {-}\textsc {vd}\) is \(\mathsf {NP}\)-complete when restricted to split graphs without \(d+1\)-claws. On the positive side, we prove that \({d}\text {-}\textsc {claw}\text {-}\textsc {vd}\) is polynomially solvable on what we call d-block graphs, a class that properly contains all block graphs.

A note on the strength and minimum color sum of bipartite graphs

The edge-closure concept for claw-free graphs and the stability of forbidden subgraphs

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We study the edge‐identifying code problem, i.e. the identifying code problem in line graphs. If denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound , where n denotes the order of G, can be improved to in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound , where is the line graph of G, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud, and the first author holds for a subclass of line graphs. Finally, we show that the edge‐identifying code problem is NP‐complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.

The Induced Graph Matching problem asks to find $$k$$ disjoint induced subgraphs isomorphic to a given graph $$H$$ in a given graph $$G$$ such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in $$k$$ on claw-free graphs when $$H$$ is a fixed connected graph, and even admits a polynomial kernel when $$H$$ is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove $$\mathsf {W}[1]$$ -hardness of Induced Graph Matching on graphs excluding $$K_{1,4}$$ as an induced subgraph, for any fixed complete graph $$H$$ . In particular, we show that Independent Set is $$\mathsf {W}[1]$$ -hard on $$K_{1,4}$$ -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or $$\mathsf {NP}$$ -complete, depending on the connectivity of $$H$$ and the structure of $$G$$ .

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

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.

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.

Maximum weight independent set for [formula omitted]claw-free graphs in polynomial time

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.&#x0D;

An hourglass is the only graph with degree sequence 4, 2, 2, 2, 2 (i.e. two triangles meeting in exactly one vertex). There are infinitely many claw-free graphs G such that G is not hamiltonian connected while its Ryjacek closure cl(G) is hamiltonian connected. This raises such a problem what conditions can guarantee that a claw-free graph G is hamiltonian connected if and only if cl(G) is hamiltonian connected. In this paper, we will do exploration toward the direction, and show that a 3-connected {claw, (P6)2, hourglass}-free graph G with minimum degree at least 4 is hamiltonian connected if and only if cl(G) is hamiltonian connected, where (P6)2 is the square of a path P6 on 6 vertices. Using the result, we prove that every 4-connected {claw, (P6)2, hourglass}-free graph is hamiltonian connected, hereby generalizing the result that every 4-connected hourglass-free line graph is hamiltonian connected by Kriesell [J Combinatorial Theory (B) 82 (2001), 306–315]. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:285-298, 2011 © 2011 Wiley Periodicals, Inc.