Centers of triangulated graphs

Covering all cliques of a graph

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

Agraph G is a diameter graph in ℝd if its vertex set is a finite subset in ℝd of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

Analysis of the structural properties of social networks has gained much interest nowadays due to its diverse range of applications. When communications between entities (i.e., edges) of a social network (graph) are stamped with time, we want to analyze all subgraphs within an arbitrary query time slice so that the number of a specific subgraph can be counted and reported quickly. We present data structures to answer such queries for triangles, quadrangles, complete subgraphs, and maximal complete subgraphs.

Assessing the stability of a clustering method involves the measurement of the extent to which the generated clusters are affected by perturbations in the input data. A measure which specifies the disturbance in a set of clusters as the minimum number of operations required to restore the set of modified clusters to the original ones is adopted. A number of well-known graph theoretic clustering methods are compared in terms of their stability as determined by this measure. Specifically, it is shown that among the clustering methods in any of several families of graph theoretic methods, clusters defined as the connected components are the most stable and the clusters specified as the maximal complete subgraphs are the least stable. Furthermore, as one proceeds from the method producing the most narrow clusters (maximal complete subgraphs) to those producing relatively broader clusters, the clustering process is shown to remain at least as stable as any method in the previous stages. Finally, the lower and the upper bounds for the measure of stability, when clusters are defined as the connected components, are derived.

The compatibility relation occurs in many different disciplines in science and engineering. When a compatibility relation exists between pairs of elements in a set, an important problem is to derive the collection of aU those elements that form maximal compatibles. If the set of elements with the compatibility relation can be visualized as a compatibility graph of which the different nodes represent the elements of the set, the only edges of the graph being the nonoriented lines joining pairs of elements with the compatibility relation, then the problem of deriving the maximal compatibles becomes identical to the graph theory problem of finding all the maximal complete subgraphs in a symmetric graph. Recently, in connection with simplifying incompletely specified sequential machines, where a kind of compatibility relation also exists between pairs of internal states, Das and Sheng proposed a method for deriving the different maximal compatibles through finding all of the modified cut-sets of the incompatibility graph of the machine. This paper, without confining itself to only incompletely specified machines, considers the problem involving the compatibility relation in a broader perspective and suggests a new approach for finding aU the modified cut-sets of the incompatibility graph of a set having a compatibility relation between its different pairs of elements.

Network biology integrates different kinds of data, including physical or functional networks and disease gene sets, to interpret human disease. A clique (maximal complete subgraph) in a protein-protein interaction network is a topological module and possesses inherently biological significance. A disease-related clique possibly associates with complex diseases. Fully identifying disease components in a clique is conductive to uncovering disease mechanisms. This paper proposes an approach of predicting disease proteins based on cliques in a protein-protein interaction network. To tolerate false positive and negative interactions in protein networks, extending cliques and scoring predicted disease proteins with gene ontology terms are introduced to the clique-based method. Precisions of predicted disease proteins are verified by disease phenotypes and steadily keep to more than 95%. The predicted disease proteins associated with cliques can partly complement mapping between genotype and phenotype, and provide clues for understanding the pathogenesis of serious diseases.

On isometric subgraphs of infinite bridged graphs and geodesic convexity

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$ . Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$ . This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$ . We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co- $H$ -space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

Finding a central vertex in an HHD-free graph

An $L( 2,1 )$-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The $L( 2,1 )$-labeling number of $G,\lambda ( G )$, is the minimum range of labels over all such labelings. It is shown that, for chordal graphs G with maximum degree $\Delta ( G ),\lambda ( G ) \leq ( \Delta ( G ) + 3 )^2 /4$; in particular, if G is a unit interval graph with chromatic number $\chi ( G ),\lambda ( G ) \leq 2\chi ( G )$, which is a better bound. As a consequence, it is shown that the conjecture $\lambda ( G ) \leq \Delta^2 ( G )$ by Griggs and Yeh [SIAM J. Discrete Math., 5 (1992), pp. 586–595] is true for chordal graphs.

Chordal graphs are triangulated graphs where any cycle larger than three is bisected by a chord. Many combinatorial optimization problems such as computing the size of the maximum clique and the chromatic number are NP-hard on general graphs but have polynomial time solutions on chordal graphs. In this paper, we present a novel multithreaded algorithm to extract a maximal chordal sub graph from a general graph. We develop an iterative approach where each thread can asynchronously update a subset of edges that are dynamically assigned to it per iteration and implement our algorithm on two different multithreaded architectures -- Cray XMT, a massively multithreaded platform, and AMD Magny-Cours, a shared memory multicore platform. In addition to the proof of correctness, we present the performance of our algorithm using a test set of synthetical graphs with up to half-a-billion edges and real world networks from gene correlation studies and demonstrate that our algorithm achieves high scalability for all inputs on both types of architectures.

The nilpotent graph is a graph’s structure motivated by the characteristics of some elements of a ring. If G is a graph, G(N) is a nilpotent graph in which the set of all non-nilpotent elements of a ring taken as a vertex set. For each x, y in R\ il(R), x and y are said to be connected or adjacent to an edge, if the sum of the vertices x and y is equal to the nilpotent element in the ring. This research aims to explain how to construct a nilpotent graph G(N) from a commutative ring. The results of this research are the characteristic of the nilpotent graph of a ring Zn for n prime numbers, a nilpotent graph for the ring Z 2 k, and ring Zn with n = 6m where 1 ≤ m < 5 can create 3 nilpotent subgraphs, they are a complete subgraph (km ) and 2 regular subgraphs in which the degree of vertices is equal to m.

Given an undirected graph G=(V,E) with vertex set V={1,?,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.

For a graph G(V,E) with vertex set V and edge set E, a set S ⊆ V is called a power dominating set (PDS), if every vertex u ∈ V - S is observed by some vertices in S using the following rules: (i) if a vertex v in G is in PDS, then it dominates itself and all the adjacent vertices of v and (ii) if an observed vertex v in G has k > 1 adjacent vertices and if k - 1 of these vertices are already observed, then the remaining one non-observed vertex is also observed by v in G. A power dominating set S ⊆ V of G (V, E) is said to be an equitable power dominating set, if for every vertex v ∈ V - S there exists an adjacent vertex u ∈ S such that the difference between degree of u and degree of v is less than or equal to 1, that is |d(u) - d(v)| ≤ 1. The minimum cardinality of an equitable power dominating set of G is called the equitable power domination number of G and is denoted by γepd (G). Let G = (V, E) be a graph with V = S1 ∪ S2 ∪ …St ∪ T where each Si is a set of vertices having cardinality of at least two and the vertices are of the same degree and T = V\ ∪ St. The degree splitting graph of G is obtained from G by adding vertices w1, w2, …, wt and joining wi to each vertex of Si (1 ≤ i ≤ t) and is denoted by DS(G). In this paper we establish the equitable power domination number of the degree splitting graph of certain graphs.