29 research outputs found

    Longest Path and Cycle Transversal and Gallai Families

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    A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|

    Recognizing H-Graphs - Beyond Circular-Arc Graphs

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    Domination and Cut Problems on Chordal Graphs with Bounded Leafage

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    The leafage of a chordal graph G is the minimum integer l such that G can berealized as an intersection graph of subtrees of a tree with l leaves. Weconsider structural parameterization by the leafage of classical domination andcut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018,Algorithmica 2020] proved, among other things, that Dominating Set on chordalgraphs admits an algorithm running in time 2O(l2)nO(1)2^{O(l^2)} n^{O(1)}. We present aconceptually much simpler algorithm that runs in time 2O(l)nO(1)2^{O(l)} n^{O(1)}. Weextend our approach to obtain similar results for Connected Dominating Set andSteiner Tree. We then consider the two classical cut problems MultiCut withUndeletable Terminals and Multiway Cut with Undeletable Terminals. We provethat the former is W[1]-hard when parameterized by the leafage and complementthis result by presenting a simple nO(l)n^{O(l)}-time algorithm. To our surprise,we find that Multiway Cut with Undeletable Terminals on chordal graphs can besolved, in contrast, in nO(1)n^{O(1)}-time.<br

    Domination and Cut Problems on Chordal Graphs with Bounded Leafage

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    The leafage of a chordal graph GG is the minimum integer \ell such that GG can be realized as an intersection graph of subtrees of a tree with \ell leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond~[ESA~20182018, Algorithmica~20202020] proved, among other things, that \textsc{Dominating Set} on chordal graphs admits an algorithm running in time 2O(2)nO(1)2^{\mathcal{O}(\ell^2)} \cdot n^{\mathcal{O}(1)}. We present a conceptually much simpler algorithm that runs in time 2O()nO(1)2^{\mathcal{O}(\ell)} \cdot n^{\mathcal{O}(1)}. We extend our approach to obtain similar results for \textsc{Connected Dominating Set} and \textsc{Steiner Tree}. We then consider the two classical cut problems \textsc{MultiCut with Undeletable Terminals} and \textsc{Multiway Cut with Undeletable Terminals}. We prove that the former is \textsf{W}[1]-hard when parameterized by the leafage and complement this result by presenting a simple nO()n^{\mathcal{O}(\ell)}-time algorithm. To our surprise, we find that \textsc{Multiway Cut with Undeletable Terminals} on chordal graphs can be solved, in contrast, in nO(1)n^{\mathcal{O}(1)}-time

    Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs

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    In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph HH, the class of HH-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of HH. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of HH-graphs for different graphs HH. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and Zeman showed, for every fixed tree TT, a polynomial-time algorithm recognizing TT-graphs. Tucker showed a polynomial time algorithm recognizing K3K_3-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of HH-graphs is NPNP-hard if HH contains two different cycles sharing an edge. The main two results of this work narrow the gap between the NPNP-hard and PP cases of HH-graphs recognition. First, we show that recognition of HH-graphs is NPNP-hard when HH contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing LL-graphs, where LL is a graph containing a cycle and an edge attached to it (LL-graphs are called lollipop graphs). Our work leaves open the recognition problems of MM-graphs for every unicyclic graph MM different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of MM-graphs, where MM is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of MM-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of MM-graphs, where MM runs over all unicyclic graphs, is NPNP-complete

    Domination and Cut Problems on Chordal Graphs with Bounded Leafage

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    Obstructions to Faster Diameter Computation: Asteroidal Sets

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    Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let ExtαExt_{\alpha} be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than α\alpha pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every mm-edge graph in ExtαExt_{\alpha} can be computed in deterministic O(α3m3/2){\cal O}(\alpha^3 m^{3/2}) time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive +1+1-approximation of all vertex eccentricities in deterministic O(α2m){\cal O}(\alpha^2 m) time. This is in sharp contrast with general mm-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in O(m2ϵ){\cal O}(m^{2-\epsilon}) time for any ϵ>0\epsilon > 0. As important special cases of our main result, we derive an O(m3/2){\cal O}(m^{3/2})-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an O(k3m3/2){\cal O}(k^3m^{3/2})-time algorithm for this problem on graphs of asteroidal number at most kk. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions

    On weighted clique graphs

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    Let K(G) be the clique graph of a graph G. A m-weighting of K(G) consists on giving to each m-size subset of its vertices a weight equal to the size of the intersection of the m corresponding cliques of G. The 2-weighted clique graph was previously considered by McKee. In this work we obtain a characterization of weighted clique graphs similar to Roberts and Spencer’s characterization for clique graphs. Some graph classes can be naturally defined in terms of their weighted clique graphs, for example clique-Helly graphs and their generalizations, and diamond-free graphs. The main contribution of this work is to characterize several graph classes by means of their weighted clique graph: hereditary clique-Helly graphs, split graphs, chordal graphs, UV graphs, interval graphs, proper interval graphs, trees, and block graphs.Sociedad Argentina de Informática e Investigación Operativ

    Graphs with at most two moplexes

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    A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: Unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of kk-moplex graphs, which are defined as graphs containing at most kk moplexes. Of particular interest is the smallest nontrivial case k=2k=2, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected 22-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected 22-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets

    The Neighborhood Polynomial of Chordal Graphs

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    The neighborhood polynomial of a graph GG is the generating function of subsets of vertices in GG that have a common neighbor. In this paper we study the neighborhood polynomial and the complexity of its computation for chordal graphs. We will show that it is \NP-hard to compute the neighborhood polynomial on general chordal graphs. Furthermore we will introduce a parameter for chordal graphs called anchor width and an algorithm to compute the neighborhood polynomial which runs in polynomial time if the anchor width is polynomially bounded. Finally we will show that we can bound the anchor width for chordal comparability graphs and chordal graphs with bounded leafage. The leafage of a chordal graphs is the minimum number of leaves in the host tree of a subtree representation. In particular, interval graphs have leafage at most 2. This shows that the anchor width of interval graphs is at most quadratic
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