57 research outputs found

    How is a Chordal Graph like a Supersolvable Binary Matroid?

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    Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.Comment: 10 pages, 3 figures, to appear in Discrete Mathematic

    On clique-colouring of graphs with few P4's

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    Abstract Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring. In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P 4's. In particular we shall consider the classes of extended P 4-laden graphs, p-trees (graphs which contain exactly n−3 P 4's) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P 4's. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P 4-reducible graphs, P 4-sparse graphs, extended P 4-reducible graphs, extended P 4-sparse graphs, P 4-extendible graphs, P 4-lite graphs, P 4-tidy graphs and P 4-laden graphs that are included in the class of extended P 4-laden graphs

    A representation for the modules of a graph and applications

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    We describe a simple representation for the modules of a graph C. We show that the modules of C are in one-to-one correspondence with the ideaIs of certain posets. These posets are characterizaded and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate alI modules of C, (ii) count the number of modules of C, (iii) find a maximal module satisfying some hereditary property of C and (iv) find a connected non-trivial module of C

    FPT algorithms to recognize well covered graphs

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    Given a graph GG, let vc(G)vc(G) and vc+(G)vc^+(G) be the sizes of a minimum and a maximum minimal vertex covers of GG, respectively. We say that GG is well covered if vc(G)=vc+(G)vc(G)=vc^+(G) (that is, all minimal vertex covers have the same size). Determining if a graph is well covered is a coNP-complete problem. In this paper, we obtain O∗(2vc)O^*(2^{vc})-time and O∗(1.4656vc+)O^*(1.4656^{vc^+})-time algorithms to decide well coveredness, improving results of Boria et. al. (2015). Moreover, using crown decomposition, we show that such problems admit kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved that recognizing well covered graphs is coW[2]-hard when the independence number α(G)=n−vc(G)\alpha(G)=n-vc(G) is the parameter. Contrasting with such coW[2]-hardness, we present an FPT algorithm to decide well coveredness when α(G)\alpha(G) and the degeneracy of the input graph GG are aggregate parameters. Finally, we use the primeval decomposition technique to obtain a linear time algorithm for extended P4P_4-laden graphs and (q,q−4)(q,q-4)-graphs, which is FPT parameterized by qq, improving results of Klein et al (2013).Comment: 15 pages, 2 figure

    New Results on Edge-coloring and Total-coloring of Split Graphs

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    A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph GG is said to be tt-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most tt. Given a graph GG, determining the smallest tt for which GG is tt-admissible, i.e. the stretch index of GG denoted by σ(G)\sigma(G), is the goal of the tt-admissibility problem. Split graphs are 33-admissible and can be partitioned into three subclasses: split graphs with σ=1,2\sigma=1, 2 or 33. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with Δ\Delta or Δ+1\Delta+1 colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, i.e., a total coloring of GG, it is conjectured that any graph can be total colored with Δ+1\Delta+1 or Δ+2\Delta+2 colors, and thus can be classified as Type 1 or Type 2. These both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with σ=2\sigma=2. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.Comment: 20 pages, 5 figure

    An algorithm for finding homogeneous pairs

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    AbstractA homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one

    Chordal (2,1) - graphs

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    A graph is said to be (k, l) if itsif its vertex set can be partitioned into k independent sets and l cliques. The class of (k,l) graphsappears as a natural generalization of split graphs. In this paper, we describe a characterization leads to a O (nm) recognition algorithm, where n and m are the numbers of vertices and edges of the input graph, respectively

    Partitioning chordal graphs into independent sets and cliques

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    We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)-graphs in general. (For instance, being a (k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem — given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minimum number of cliques that meet all Kr's
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