1,263 research outputs found
Model Counting for Formulas of Bounded Clique-Width
We show that #SAT is polynomial-time tractable for classes of CNF formulas
whose incidence graphs have bounded symmetric clique-width (or bounded
clique-width, or bounded rank-width). This result strictly generalizes
polynomial-time tractability results for classes of formulas with signed
incidence graphs of bounded clique-width and classes of formulas with incidence
graphs of bounded modular treewidth, which were the most general results of
this kind known so far.Comment: Extended version of a paper published at ISAAC 201
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
Multi-Clique-Width
Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorability. In particular, c-colorability can be tested in time linear in n and singly exponential in c and the width k of a given multi-k-expression. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs. This gap shows up when the tree-width is basically equal to the multi-clique width as well as when the tree-width is not bounded by any function of the clique-width
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
- …