Showing papers on "K-tree published in 2017"

Enumeration of Maximal Cliques from an Uncertain Graph

Abstract: We consider the enumeration of dense substructures (maximal cliques) from an uncertain graph. For parameter 0 <; α <; 1, we define the notion of an a-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of a-maximal cliques possible within a (uncertain) graph. We present an algorithm to enumerate a-maximal cliques whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.

Mining Maximal Cliques on Dynamic Graphs Efficiently by Local Strategies

Abstract: Maximal Clique Enumeration (MCE) is a long standing problem in database community. Though it is extensively studied, almost all researches focus on calculating maximal cliques as a one-time effort. MCE on dynamic graph has been rarely discussed so far, the only work on this topic is to maintain maximal cliques with graph evolving. The key within this problem is to find maximal cliques that contains vertices incident to the inserted edge when edge insertion happens. Up to O(W2) candidates are generated in prior method based on Cartesian product, the overall complexity is O(W2n2) where n, W represents the number of vertices and maximal cliques on the graph. Besides, maximality verification of candidate is conducted frequently by global search. Change of maximal clique induced by graph's updating presents some localities. We propose novel local construction strategy to generate candidates based on linear scan, number of candidates is reduced to O(W), the overall complexity is then reduced to O(Wn2). Furthermore, we present heuristics to reduce the cost incurred by maximality verification. Theoretical analysis and experiments on real graphs indicate that our proposals are effective and efficient.

Efficient Enumeration of Maximal k-Degenerate Subgraphs in a Chordal Graph

Abstract: In this paper, we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay \(O(m\cdot \omega (G))\) for any n-vertex chordal graph with m edges, where \(\omega (G) \le n\) is the maximum size of a clique in G. The problem generalizes that of enumerating maximal independent sets and maximal induced forests, which correspond to respectively 0-degenerate and 1-degenerate subgraphs.

A characterization of some graphs with metric dimension two

Abstract: A set W ⊆ V (G) is called a resolving set, if for each pair of distinct vertices u,v ∈ V (G) there exists t ∈ W such that d(u,t)≠d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dimM(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) =deg(v) = j. In this paper, we prove that if G is a k-path, then dimM(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two.

A linear-time algorithm for clique-coloring problem in circular-arc graphs

Abstract: A maximal clique of G is a clique not properly contained in any other clique. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no maximal clique with at least two vertices is monochromatic. The smallest integer k admitting a k-clique-coloring of G is called clique-coloring number of G. Cerioli and Korenchendler (Electron Notes Discret Math 35:287---292, 2009) showed that there is a polynomial-time algorithm to solve the clique-coloring problem in circular-arc graphs and asked whether there exists a linear-time algorithm to find an optimal clique-coloring in circular-arc graphs or not. In this paper we present a linear-time algorithm of the optimal clique-coloring in circular-arc graphs.

On Maximal Cliques with Connectivity Constraints in Directed Graphs

Abstract: Finding communities in the form of cohesive subgraphs is a fundamental problem in network analysis. In domains that model networks as undirected graphs, communities are generally associated with dense subgraphs, and many community models have been proposed. Maximal cliques are arguably the most widely studied among such models, with early works dating back to the '60s, and a continuous stream of research up to the present. In domains that model networks as directed graphs, several approaches for community detection have been proposed, but there seems to be no clear model of cohesive subgraph, i.e., of what a community should look like. We extend the fundamental model of clique to directed graphs, adding the natural constraint of strong connectivity within the clique. We characterize the problem by giving a tight bound for the number of such cliques in a graph, and highlighting useful structural properties. We then exploit these properties to produce the first algorithm with polynomial delay for enumerating maximal strongly connected cliques.

Large-scale graph data processing method based on k -tree and multi-valued decision diagram

Abstract: The invention discloses a large-scale graph data processing method based on a k -tree and a Multi-valued Decision Diagram (MDD). The method comprises the following steps: 1, performing n-bit encoding on a peak of a graph according to the rule of the k -tree, FORMULA, k being greater than or equal to 2; 2, encoding an edge according to a peak code; 3, constructing an MDD structure according to an edge code to obtain a k -MDD structure which corresponds to an oriented graph and contains n variables, wherein the k -MDD structure has the nature of the MDD and is applicable to the simplification rule of the MDD; 4, performing the following basic operation of the graph on the obtained k -MDD structure through logic operation of a symbol decision diagram: edge inquiry, outside adjacent inquiry and calculation of an out degree of the peak, inside adjacent inquiry and calculation of an in degree of the peak, as well as adding and deletion of an edge, and the like. According to the method, the MDD is used for storing graph data, so that isomorphic sub trees in the k -tree are combined, the number of nodes is reduced, and the structure is more compact; the basic operation of the graph is converted into the logic operation, so that the operation is simpler.

Reconstructing Nearly Simple Polytopes from their Graph

Abstract: We present a partial description of which polytopes are reconstructible from their graphs. This is an extension of work by Blind and Mani (1987) and Kalai (1988), which showed that simple polytopes can be reconstructed from their graphs. In particular, we introduce a notion of $h$-nearly simple and prove that 1-nearly simple and 2-nearly simple polytopes are reconstructible from their graphs. We also give an example of a 3-nearly simple polytope which is not reconstructible from its graph. Furthermore, we give a partial list of polytopes which are reconstructible from their graphs in an entirely non-constructive way.

Computing a Clique Tree with the Algorithm Maximal Label Search

Abstract: The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.

Detecting Bases of Maximal Cliques in a Graph

Abstract: Maximal Cliques Enumeration (MCE), as a fundamental problem, has been extensively investigated in many fields, such as social networks, and biological science and so forth. However, the existing research works usually ignore the formation principle of maximal cliques which can help us to speed up the detection of maximal cliques in a graph. This paper pioneers a novel problem on detection of bases of maximal cliques in a graph. We propose a formal concept analysis based approach for detecting the bases of maximal cliques and detection theorem. It is believed that our work can provide a new research solution and direction for future topological structure analysis in various complex networking systems.

Independence Number and k-Trees of Graphs

Abstract: A tree T is called a k-tree if the maximum degree of T is at most k. In this paper, we give a sufficient condition for a graph to have a k-tree containing specified vertices as following: let G be a connected graph and let S be a subset of V(G). If \(\alpha _G(S)\le (k-1)\kappa _G(S)+1\), then G has a k-tree containing S. Moreover, this condition is sharp.

A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes

Abstract: ・ Reversibility and Foldability of Conway Tiles (with K. Matsunaga), Computational Geometry, Vol.64 (2017), 30-45 ・ A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes (with Sin Hitotumatu, Motonaga Ishii, Akihiro Matuura, Ikuro Sato and Shun Toyoshima), Journal of Information Processing Vol.25, 528-536 (2017) ・ Generalization of Haberdasher’s Puzzle (with Kiyoko Matsunaga), Discrete and Computational Geometry Vol.58 Issue 1, 30-50 ・ Reversible Nets of Polyhedra (with S. Langerman and K. Matsunaga), Discrete and Computational Geometry and Graphs 2015, LNCS9943 (2016),13-23

Algorithms for k-Internal Out-Branching and k-Tree in Bounded Degree Graphs

Abstract: In this paper, we employ the multilinear detection technique, combined with proper colorings of graphs, to develop algorithms for two problems in bounded degree graphs. We focus mostly on the k-Internal Out-Branching (k-IOB) problem, which asks if a given directed graph has an out-branching (i.e., a spanning tree with exactly one node of in-degree 0) with at least k internal nodes. The second problem, k-Tree, asks if a given undirected graph G has a (not necessarily induced) copy of a given tree T. That is, k-Tree asks whether T is a subgraph of G. We present an $$O^*(4^k)$$Oź(4k) time randomized algorithm for k-IOB, which improves the $$O^*$$Oź running time of the previous best known algorithm for this problem. Then, for directed graphs whose underlying (simple, undirected) graphs have bounded degree $$\varDelta $$Δ, we modify our algorithm to solve k-IOB in time $$O^*(2^{(2-\frac{\varDelta +1}{\varDelta (\varDelta -1)})k})$$Oź(2(2-Δ+1Δ(Δ-1))k). For k- Tree in graphs of bounded degree 3, we obtain an $$O^*(1.914^k)$$Oź(1.914k) time randomized algorithm. In particular, all of our algorithms use polynomial space.

Graph Reconstruction in the Congested Clique

Abstract: The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of $n$ nodes. In this paper we consider the situation where the joint input to the nodes is an $n$-node labeled graph $G$, i.e., the local input received by each node is the indicator function of its neighborhood in $G$. Nodes execute an algorithm, communicating with each other in synchronous rounds and their goal is to compute some function that depends on $G$. In every round, each of the $n$ nodes may send up to $n-1$ different $b$-bit messages through each of its $n-1$ communication links. We denote by $R$ the number of rounds of the algorithm. The product $Rb$, that is, the total number of bits received by a node through one link, is the cost of the algorithm. The most difficult problem we could attempt to solve is the reconstruction problem, where nodes are asked to recover all the edges of the input graph $G$. Formally, given a class of graphs $\mathcal G$, the problem is defined as follows: if $G otin {\mathcal G}$, then every node must reject; on the other hand, if $G \in {\mathcal G}$, then every node must end up, after the $R$ rounds, knowing all the edges of $G$. It is not difficult to see that the cost $Rb$ of any algorithm that solves this problem (even with public coins) is at least $\Omega(\log|\mathcal{G}_n|/n)$, where $\mathcal{G}_n$ is the subclass of all $n$-node labeled graphs in $\mathcal G$. In this paper we prove that previous bound is tight and that it is possible to achieve it with only $R=2$ rounds. More precisely, we exhibit (i) a one-round algorithm that achieves this bound for hereditary graph classes; and (ii) a two-round algorithm that achieves this bound for arbitrary graph classes.

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

Abstract: Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex transitive, not even regular.

Affine maps between quadratic assignment polytopes and subgraph isomorphism polytopes

Abstract: We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P((n-2,2))$. He also posed the question whether $QAP(n)$ and $P((n-2,2))$, having $n!$ vertices each, are isomorphic. We show that $QAP(n)$ and $P((n-2,2))$ are not isomorphic. Also, we show that $QAP(n)$ is a face of $P((2n-2,2))$, but $P((n-2,2))$ is a projection of $QAP(n)$.

Reflexive polytopes arising from partially ordered sets and perfect graphs

Abstract: Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Grobner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $\delta$-polynomials of these reflexive polytopes.

Extremal properties of distance-based graph invariants for $k$-trees

Abstract: Sharp bounds on some distance-based graph invariants of $n$-vertex $k$-trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among $k$-trees with $n$ vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second maximal reciprocal product-degree distance (and similarly, the extremal graphs with the minimal and the second minimal degree distance are coincident with graphs having the minimal and the second minimal eccentricity distance sum).