New width parameters of graphs (original) (raw)
Related papers
Discrete Applied Mathematics, 2016
dedicated to the second workshop in the series, held in 2005 in Prague, Czech Republic; 145-2 (2005) dedicated to the first workshop, held in 2001 in Barcelona, Spain; and 54-2/3 (1994) dedicated to a workshop held in 1989 in Eugene, which in retrospect we view as workshop number zero in what has evolved to become the successful biannual GROW workshop series. This issue comprises 14 papers authored mainly, but not exclusively, by participants of the workshop. All submissions have been carefully refereed, and we thank all the referees for their hard work. True to the name of the workshop, the papers in the current special issue report on investigations in three areas of research: Graph classes, Optimization, and Width parameters. Due to the close interconnections among these areas, most of the papers fit into more than one of them. Based on their main focus, we introduce the papers in this issue in the corresponding three groups. The area of Graph Classes is represented by papers proving new structural properties of various graph classes and exploring algorithmic consequences of these properties. Bonomo, Grippo, Milanič, and Safe initiate the study of graph classes of power-bounded clique-width and give sufficient and necessary conditions for this property. Brignall, Lozin, and Stacho study bichain graphs that are a bipartite analog of split permutation graphs. They show that these graphs admit a simple geometric representation and have a universal element of quadratic order. Golovach, Heggernes, Kanté, Kratsch, and Villanger show that all minimal dominating sets of a chordal bipartite graph can be generated in incremental polynomial, hence output polynomial, time. Konagaya, Otachi, and Uehara present a polynomial-time algorithm for the subgraph isomorphism problem on several subclasses of perfect graphs. Mertzios and Zaks study a conjecture by Golumbic, Monma, and Trotter that states that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. They prove the conjecture for every graph that admits a tolerance representation with exactly one unbounded vertex. The Optimization section consists of papers that study computational complexity and algorithmic issues of various optimization problems on graphs. Corneil, Dusart, Habib, Mamcarz, and de Montgolfier consider the problem of the recognition of various kinds of orderings produced by graph searches. To this aim, they introduce a new framework in order to handle a broad variety of
Lecture Notes in Computer Science, 2009
We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divide-and-conquer approach. Boolean-width is similar to rank-width, which is related to the number of GF [2]-sums (1+1=0) of neighborhoods instead of the Boolean-sums (1+1=1) used for boolean-width. For an n-vertex graph G given with a decomposition tree of boolean-width k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n + 2 3k k)). We show that for any graph the square root of its boolean-width is never more than its rank-width. We also exhibit a class of graphs, the Hsugrids, having the property that a Hsu-grid on Θ(n 2) vertices has boolean-width Θ(log n) and tree-width, branch-width, clique-width and rank-width Θ(n). Moreover, any optimal rank-decomposition of such a graph will have boolean-width Θ(n) , i.e. exponential in the optimal boolean-width.
On the Boolean-Width of a Graph: Structure and Applications
Lecture Notes in Computer Science, 2010
Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of booleanwidth k , we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O * (2 O(k 2)). We relate the boolean-width of a graph to its branch-width and to the booleanwidth of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width by Oum (JGT 2008). For a random graph on n vertices we show that almost surely its boolean-width is Θ(log 2 n)-setting boolean-width apart from other graph invariants-and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time O * (2 O(log 4 n)) .
On classes of graphs with logarithmic boolean-width
2016
Boolean-width is a recently introduced graph parameter. Many problems are fixed param-eter tractable when parametrized by boolean-width, for instance "Minimum Weighted Dom-inating Set " (MWDS) problem can be solved in O∗(23k) time given a boolean-decomposition of width k, hence for all graph classes where a boolean-decomposition of width O(log n) can be found in polynomial time, MWDS can be solved in polynomial time. We study graph classes having boolean-width O(log n) and problems solvable in O∗(2O(k)), combining these two results to design polynomial algorithms. We show that for trapezoid graphs, circular permutation graphs, convex graphs, Dilworth-k graphs, circular arc graphs and complements of k-degenerate graphs, boolean-decompositions of width O(log n) can be found in polyno-mial time. We also show that circular k-trapezoid graphs have boolean-width O(log n), and find such a decomposition if a circular k-trapezoid intersection model is given. For many of the graph c...
Graph Classes with Structured Neighborhoods and
2012
Given a graph in any of the following graph classes: trapezoid graphs, circular permutation graphs, convex graphs, Dilworth k graphs, k-polygon graphs, circular arc graphs and complements of k-degenerate graphs, we show how to compute decompositions with the number of d-neighborhoods bounded by a polynomial of the input size. Combined with results of Bui-Xuan, Telle and Vatshelle [1] this leads to polynomial time algorithms for a large class of locally checkable vertex subset and vertex partitioning problems on all of these graph classes. The boolean-width of a graph is related to the number of 1-neighbourhoods and our results imply that any of these graph classes have boolean-width O(logn).
On the threshold-width of graphs
Journal of Graph Algorithms and Applications, 2011
For a graph class G, a graph G has G-width k if there are k independent sets N1, . . . , N k in G such that G can be embedded into a graph H ∈ G such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k
Exploiting graph structure to cope with hard problems (Dagstuhl Seminar 11182)
This report documents the program and the outcomes of Dagstuhl Seminar 11182 "Exploiting graph structure to cope with hard problems" which has been held in Schloss Dagstuhl -Leibniz Center for Informatics from May 1st, 2011 to May 6th, 2011. During the seminar experts with a common focus on graph algorithms presented various new results in how to attack NP-hard graph problems by using the structure of the input graph. Moreover, in the afternoon of each seminar's day new problems have been posed and discussed.
Exploiting restricted linear structure to cope with the hardness of clique-width
2010
Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant linear clique-width, in polynomial time. In TAMC 2009, a polynomialtime algorithm for computing linear clique-width on a subclass of proper interval graphs was given. In this paper, we present a polynomial-time algorithm for a larger subclass of proper interval graphs that approximates the clique-width within an additive factor 3. Previously known upper bounds on clique-width result in arbitrarily large difference from the actual clique-width when applied on this class. Our results contribute toward the goal of eventually obtaining a polynomial-time exact algorithm for clique-width on proper interval graphs.
2009
The mathcalG\mathcal{G}mathcalG -width of a class of graphs mathcalG\mathcal{G}mathcalG is defined as follows. A graph G has mathcalG\mathcal{G}mathcalG -width k if there are k independent sets mathbbN1,dots,mathbbNrmttk\mathbb{N}_1,\dots,\mathbb{N}_{\rm \tt k}mathbbN1,dots,mathbbNrmttk in G such that G can be embedded into a graph rmHinmathcalG{\rm H \in \mathcal{G}}rmHinmathcalG such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in ℕi. For the class mathfrakB\mathfrak{B}mathfrakB of block graphs we show that mathfrakB\mathfrak{B}mathfrakB -width is NP-complete and we present fixed-parameter algorithms.