Bounded Clique Width Research Articles

Vertex coloring (4K1, hole-twin, 5-wheel)-free graphs

There has been recently keen interest in finding polynomial-time algorithms for the VERTEX COLORING problem for graphs G that are F-free for a given set F of graphs. If L is a set of four-vertex graphs, then the complexity of VERTEX COLORING for L-free graphs is known with three exceptions: L1 = {claw, 4K1}, L2 = {claw, 4K1, co-diamond}, and L3 = {4K1, C4}. In this paper, we study a problem arising from the class L3. A hole is an induced cycle with at least four vertices. A hole-twin is the graph obtained from a hole by adding a vertex that forms true twins with some vertex of the hole. A 5-wheel is the graph obtained from a C5 by adding a vertex that is adjacent to all vertices of the C5. We prove that a (4K1, hole-twin, 5-wheel)-free graph is perfect or has bounded clique width. As consequence we obtain a polynomial time algorithm for VERTEX COLORING for (4K1, hole-twin, 5-wheel)-free graphs.

Graph Isomorphism for (H_1,H_2)-Free Graphs: An\xa0Almost Complete Dichotomy

We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs H_{1} and H_2 for all but six pairs (H_1,H_2). Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for (H_1,H_2)-free graphs to five.

Parameterized complexity of distance labeling and uniform channel assignment problems

We rephrase the Distance labeling problem as a specific uniform variant of the Channel Assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the Distance labeling problem is FPT when parameterized by the neighborhood diversity, the maximum pi and k. This is indeed a more general answer to an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover.Finally, we show that the uniform variant of the Channel Assignment problem becomes NP-complete when generalized to graphs of bounded clique width.

Characterizations of ([formula omitted], [formula omitted], [formula omitted])-free graphs

Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. There has been keen interest in colouring graphs whose forbidden list L contains graphs with four vertices. A recent paper of Lozin and Malyshev (in press) discusses the state of the art on this problem, identifying three outstanding classes: L=(4K1,claw), L=(4K1,claw, co-diamond), and L=(4K1,C4). In this paper we investigate the class of (4K1, C4, C5)-free graphs and show that if G is a (4K1, C4, C5)-free graph, then G either has bounded clique width or is perfect.

Exact Algorithms for a Loading Problem with Bounded Clique Width

In this paper we discuss a special pallet-loading problem, which we encountered at a manufacturing company. In graph-theoretical terms, the problem is equivalent to partitioning a permutation graph into bounded-size cliques. We formulate the problem as an integer program, and present two exact algorithms for solving it. The first algorithm is a branch-and-price algorithm based on the integer-programming formulation; the second one is an algorithm based on the concept of bounded clique width. The latter algorithm was motivated by the structure present in the real-world instances. Test results are given, both for real-world instances and randomly generated instances. As far as we are aware, this is the first implementation of an algorithm based on bounded clique width.

On the structure of ( [formula omitted],gem)-free graphs

We give a complete structure description of ( P 5 ,gem)-free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a , b , c , d , e and four edges ab , bc , cd , de , and the gem consists of a P 4 a , b , c , d with edges ab , bc , cd plus a universal vertex e adjacent to a , b , c , d .

On the structure and stability number of P5- and co-chair-free graphs

We give a O(nm) time algorithm for the maximum weight stable set (MWS) problem on P 5- and co-chair-free graphs without recognizing whether the (arbitrary) input graph is P 5- and co-chair-free. This algorithm is based on the fact that prime P 5- and co-chair-free graphs containing 2 K 2 are matched co-bipartite graphs and thus have very simple structure, and for 2 K 2-free graphs, there is a polynomial time algorithm for the MWS problem due to a result of Farber saying that 2 K 2-free graphs contain at most O(n 2) maximal stable sets. A similar argument can be used for ( P 5,co-P)-free graphs; their prime graphs are 2 K 2-free. Moreover, we give a complete classification of ( P 5,co-chair, H)-free graphs with respect to their clique width when H is a one-vertex P 4 extension; this extends the characterization of ( P 5, P 5 ,co-chair)-free graphs called semi- P 4- sparse by Fouquet and Giakoumakis. For H being a house, P, bull or gem, the class of ( P 5,co-chair, H)-free graphs has bounded clique width and very simple structure whereas for the other four cases, namely H being a co-gem, chair, co- P or C 5, the class has unbounded clique width due to a result of Makowsky and Rotics. Bounded clique width implies linear time algorithms for all algorithmic problems expressible in LinEMSOL—a variant of Monadic Second Order Logic including the MWS Problem.

( P5,diamond)-free graphs revisited: structure and linear time optimization

Using the concept of prime graphs and modular decomposition of graphs, we give a complete structure description of ( P 5,diamond)-free graphs implying that these graphs have bounded clique width (the P 5 is the induced path with five vertices a, b, c, d, e and four edges ab, bc, cd, de, and the diamond consists of four vertices a, b, c, d such that a, b, c form an induced path with edges ab, bc, and vertex d is adjacent to a, b and c). The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is, roughly speaking, expressible in a certain kind of Monadic Second-Order Logic using quantification only over vertex but not over edge set predicates. This improves previous results on ( P 5,diamond)-free graphs in several ways: We give a complete structure description of prime ( P 5,diamond)-free graphs, we do not only solve the MWS problem on this class, we achieve linear time algorithms (instead of a recent time bound O ( n m ) ), and we can do all this on a larger graph class containing ( P 5,diamond)-free graphs which admits linear time recognition.

Structure and stability number of chair-, co-P- and gem-free graphs revisited

The P 4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b ( a and b, a, b, c and d, respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O( n 4) time bound for Maximum Stable Set on butterfly-, chair-, co-P- and gem-free graphs.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Minty has shown that the Maximum Weight Stable Set (MWS) Problem can be solved in polynomial time when restricted to claw-free graphs. We show that the structure of graphs being both claw-free and co-claw-free is very simple which implies bounded clique width for this graph class. It is known that for graph classes of bounded clique width (assuming that k-expressions can be determined in linear time), there are linear time algorithms for all problems expressible in Monadic Second Order Logic with quantification only over vertex sets. The problems MWS, Maximum Clique, Domination and Steiner Tree, for example, are expressible in this way. Moreover, we describe the structure of prime graphs being both H- and H -free for any four-vertex graph H and obtain bounded clique width for these graph classes except the 2K 2 - and C 4 -free graphs.

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantification. Such quantifications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this affects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL definable graph properties. Finally, our results are also applicable to SAT and ♯ SAT.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.