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Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$ by using only edge contractions. In this case the problem is known to be NP-complete even if $k=4$. This led to an intensive investigation for testing contractibility on restricted graph classes. We focus on bipartite graphs. Heggernes, van 't Hof, L\'{e}v\^{e}que and Paul proved that the problem stays NP-complete for bipartite graphs if $k=6$. We strengthen their result from $k=6$ to $k=5$. We also show that the problem of contracting a bipartite graph to the $6$-vertex cycle $C_6$ is NP-complete. The cyclicity of a graph is the length of the longest cycle the graph can be contracted to. As a consequence of our second result, determining the cyclicity of a bipartite graph is NP-hard., Comment: 9 pages, 2 figures

For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H.

The Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very limited number of other graph classes. By researching boundedness of the treewidth of a graph, we prove that Square Root is polynomial-time solvable on various graph classes of low clique number that are not minor-closed.

Let GG be a bipartite graph, and let HH be a bipartite graph with a fixed bipartition (BH,WH)(BH,WH). We consider three different, natural ways of forbidding HH as an induced subgraph in GG. First, GG is HH-free if it does not contain HH as an induced subgraph. Second, GG is strongly HH-free if no bipartition of GG contains an induced copy of HH in a way that respects the bipartition of HH. Third, GG is weakly HH-free if GG has at least one bipartition that does not contain an induced copy of HH in a way that respects the bipartition of HH. Lozin and Volz characterized all bipartite graphs HH for which the class of strongly HH-free bipartite graphs has bounded clique-width. We extend their result by giving complete classifications for the other two variants of HH-freeness.

The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer ℓ for which an input graph can be contracted to the path Pℓ on ℓ vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to Pℓ-free graphs jumps from being polynomially solvable to being NP-hard at ℓ=6, while this jump occurs at ℓ=5 for the 2-Disjoint Connected Subgraphs problem. We also present an exact algorithm that solves the 2-Disjoint Connected Subgraphs problem faster than O∗(2n) for any n-vertex Pℓ-free graph. For ℓ=6, its running time is O∗(1.5790n). We modify this algorithm to solve the Longest Path Contractibility problem for P6-free graphs in O∗(1.5790n) time.

Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain is a subset of $V$ that induces a maximal connected subgraph of $G$ in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$, all $k$-colourings of $k$-regular graphs that are not complete are Kempe equivalent. We address the case $k=3$ by showing that all $3$-colourings of a cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism., very minor changes

A parallel knock-out scheme for a graph proceeds in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is KO-reducible if there exists such a scheme that eliminates every vertex in the graph. The Parallel Knock-Out problem is to decide whether a graph GG is KO-reducible. This problem is known to be NP-complete and has been studied for several graph classes. We show that the problem is NP-complete even for split graphs, a subclass of P5P5-free graphs. In contrast, our main result is that it is linear-time solvable for P4P4-free graphs (cographs).

If a graph G contains no subgraph isomorphic to some graph H , then G is called H -free. A coloring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that no two adjacent vertices have the same color, i.e., c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k -coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k . The Precoloring Extension problem is to decide whether a partial k -coloring of a graph can be extended to a k -coloring of the whole graph for some integer k . The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound ℓ on the size of each L(u) we obtain the ℓ -List Coloring problem. We first classify the Precoloring Extension problem and the ℓ -List Coloring problem for H -free graphs. We then show that 3-List Coloring is NP-complete for n -vertex graphs of minimum degree n−2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n−2. Finally, for a fixed integer k>0, the Listk -Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,…,k}. We show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of Listk -Coloring for P6-free graphs.

For a set SS of graphs, a perfect SS-packing (SS-factor) of a graph GG is a set of mutually vertex-disjoint subgraphs of GG that each are isomorphic to a member of SS and that together contain all vertices of GG. If GG allows a covering (locally bijective homomorphism) to a graph HH, i.e., a vertex mapping f:VG→VHf:VG→VH satisfying the property that f(u)f(v)f(u)f(v) belongs to EHEH whenever the edge uvuv belongs to EGEG such that for every u∈VGu∈VG the restriction of ff to the neighborhood of uu is bijective, then GG is an HH-cover. For some fixed HH let S(H)S(H) consist of all connected HH-covers. Let Kk,ℓKk,ℓ be the complete bipartite graph with partition classes of size kk and ℓℓ, respectively. For all fixed k,ℓ≥1k,ℓ≥1, we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S(Kk,ℓ)S(Kk,ℓ)-packing. Our technique is partially based on exploring a close relationship to pseudo-coverings. A pseudo-covering from a graph GG to a graph HH is a homomorphism from GG to HH that becomes a covering to HH when restricted to a spanning subgraph of GG. We settle the computational complexity of the problem that asks whether a graph allows a pseudo-covering to Kk,ℓKk,ℓ for all fixed k,ℓ≥1k,ℓ≥1.

A list assignment of a graph G = ( V , E ) is a function L that assigns a list L ( u ) of so-called admissible colors to each u � V . The List Coloring problem is that of testing whether a given graph G = ( V , E ) has a coloring c that respects a given list assignment L , i.e., whether G has a mapping c : V � { 1 , 2 , � } such that (i) c ( u ) � c ( v ) whenever u v � E and (ii) c ( u ) � L ( u ) for all u � V . If a graph G has no induced subgraph isomorphic to some graph of a pair { H 1 , H 2 } , then G is called ( H 1 , H 2 ) -free. We completely characterize the complexity of List Coloring for ( H 1 , H 2 ) -free graphs.

A colouring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k -colouring. The Colouring problem is that of testing whether a given graph has a k -colouring for some given integer k . If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with |H|=1 has been completely classified. When |H|=2, the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H1,H2}, where we allow H1 to have a single edge and H2 to have a single non-edge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H1|+|H2|. As a by-product, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.

International audience; We characterize all graphs that have carving-width at most k for k=1,2,3. In particular, we show that a graph has carving-width at most 3 if and only if it has maximum degree at most 3 and treewidth at most 2. This enables us to identify the immersion obstruction set for graphs of carving-width at most 3.

The Degree Contractibility problem is to test whether a given graph G can be modified to a graph of minimum degree at least d by using at most k contractions. We prove the following three results. First, Degree Contractibility is NP -complete even when d = 14 . Second, it is fixed-parameter tractable when parameterized by k and d . Third, it is W [ 1 ] -hard when parameterized by k . We also study its variant where the input graph is weighted, i.e., has some edge weighting and the contractions preserve these weights. The Weighted Degree Contractibility problem is to test if a weighted graph G can be contracted to a weighted graph of minimum weighted degree at least d by using at most k weighted contractions. We show that this problem is NP -complete and that it is fixed-parameter tractable when parameterized by k . In addition, we pinpoint a relationship with the problem of finding a minimal edge-cut of maximum size in a graph and study the parameterized complexity of this problem and its variants.

We show that the Satisfiability (SAT) problem for CNF formulas with {\beta}-acyclic hypergraphs can be solved in polynomial time by using a special type of Davis-Putnam resolution in which each resolvent is a subset of a parent clause. We extend this class to CNF formulas for which this type of Davis-Putnam resolution still applies and show that testing membership in this class is NP-complete. We compare the class of {\beta}-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" {\beta}-acyclic instances, using as parameter the formula's distance from being {\beta}-acyclic. As distance we use the size of a smallest strong backdoor set and the {\beta}-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve., Comment: Extended abstracts appeared in the Proceedings of FSTTCS 2010 and SAT 2011. The latter corresponds to revision 1 of this arXiv paper (arXiv:1104.4279v1)

For a non-hamiltonian claw-free graph $G$ with order $n$ and minimum degree $\delta$ we show the following. If $\delta=4$, then $G$ has a 2-factor with at most $(5n-14)/18$ components, unless $G$ belongs to a finite class of exceptional graphs. If $\delta \ge 5$, then $G$ has a 2-factor with at most $(n-3)/(\delta -1)$ components. These bounds are sharp in the sense that we can replace nor 5/18 by a smaller quotient nor $\delta -1$ by $\delta$.

Consider a soccer competition among various teams playing against each other in pairs (matches) according to a previously determined schedule. At some stage of the competition one may ask whether a particular team still has a (theoretical) chance to win the competition. The complexity of this question depends on the way scores are allocated according to the outcome of a match. For example, the problem is polynomially solvable for the ancient FIFA rules (2 : 0 resp. 1 : 1) but becomes NP-hard if the new rules (3 : 0 resp. 1 : 1) are applied. We determine the complexity of the above problem for all possible score allocation rules.