Trivially-Perfect Width (original) (raw)
Related papers
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
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.
On the Clique-Width of Some Perfect Graph Classes
International Journal of Foundations of Computer Science, 2000
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.
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 the Complete Width and Edge Clique Cover Problems
Lecture Notes in Computer Science, 2015
A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V, E), the complete width of G is the minimum k such that there exist k independent sets N i ⊆ V , 1 ≤ i ≤ k, such that the graph G ′ obtained from G by adding some new edges between certain vertices inside the sets N i , 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on 3K 2-free bipartite graphs and polynomially solvable on 2K 2-free bipartite graphs and on (2K 2 , C 4)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on 3K 2-free co-bipartite graphs and polynomially solvable on C 4-free co-bipartite graphs and on (2K 2 , C 4)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most 2 k vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most 2 k vertices. Finally we determine all graphs of small complete width k ≤ 3.
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
On the structure of graphs with path-width at most two
2012
Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with pathwidth at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.
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...
On the Clique-Width of Graphs with Few P
1999
Babel and Olariu (1995) introduced the class of (q;t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be de ned by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique{width of the (q;t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q;q ? 3) graph for q 7, has clique{width q and a q{expression de ning it can be obtained in linear time. On the other hand we show that the class of (q;q ?3) graphs for 4 q 6 and the class of (q; q ?1) graphs for q 4 are not of bounded clique{width.
Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs
Discrete Applied Mathematics, 2012
We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.