Hamiltonicity of Simplicial-Connected Graphs: An Algorithm Based on Clique Decomposition

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).

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k-Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k-Subgraph is ????-hard and even not approximable unless ?????? in general graphs. Thus, we investigate Sparsest k-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the ????-hardness of Sparsest k-Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a PTAS for Sparsest k-Subgraph on proper interval graphs.

<abstract> Let $ G $ be a finite group. The reduced power graph of $ G $ is the undirected graph whose vertex set consists of all elements of $ G $, and two distinct vertices $ x $ and $ y $ are adjacent if either $ \langle x\rangle \subset \langle y\rangle $ or $ \langle y\rangle \subset \langle x\rangle $. In this paper, we show that the reduced power graph of $ G $ is perfect and characterize all finite groups whose reduced power graphs are split graphs, cographs, chordal graphs, and threshold graphs. We also give complete classifications in the case of abelian groups, dihedral groups, and generalized quaternion groups. </abstract>

For a finite undirected graph G = (V ,E ) and positive integer k *** 1, an edge set M *** E is a distance-k matching if the mutual distance of edges in M is at least k in G . For k = 1, this gives the usual notion of matching in graphs, and for general k *** 1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k = 2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G ) by Cameron and strong matching by Golumbic and Laskar in various papers. Finding a maximum induced matching is $\mathbb{NP}$-complete even on very restricted bipartite graphs but for k = 2, it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L (G )2 of the line graph L (G ) of G which, by a result of Cameron, is chordal for any chordal graph G . We show that, unlike for k = 2, for a chordal graph G , L (G )3 is not necessarily chordal, and finding a maximum distance-3 matching remains $\mathbb{NP}$-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-3 matching problem can be solved in polynomial time. Moreover, we obtain various new results for induced matchings.

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

The independent set polynomial of a graph has one variable for each vertex and one monomial for each independent set, comprising the product of the corresponding variables. Given a graph G on n vertices and a vector p ∈ [0,1)ⁿ, a central problem in statistical mechanics is determining whether the independent set polynomial of G is non-vanishing in the polydisk of p, i.e., whether |Z_G(x)| &gt; 0 for every x ∈ ℂⁿ such that |x_i| ≤ p_i. Remarkably, when this holds, Z_G(-p) is a lower bound for the avoidance probability when G is a dependency graph for n events whose probabilities form vector p. A local sufficient condition for |Z_G| &gt; 0 in the polydisk of p is the Lovász Local Lemma (LLL). &#13;\nIn this work we derive several new results on the efficient evaluation and bounding of Z_G. Our starting point is a monotone mapping from subgraphs of G to truncations of the tree of self-avoiding walks of G. Using this mapping our first result is a local upper bound for Z(-p), similar in spirit to the local lower bound for Z(-p) provided by the LLL. Next, using this mapping, we show that when G is chordal, Z_G can be computed exactly and in linear time on the entire complex plane, implying perfect sampling for the hard-core model on chordal graphs. We also revisit the task of bounding Z(-p) from below, i.e., the LLL setting, and derive four new lower bounds of increasing sophistication. Already our simplest (and weakest) bound yields a strict improvement of the famous asymmetric LLL, i.e., a strict relaxation of the inequalities of the asymmetric LLL without any further assumptions. This new asymmetric local lemma is sharp enough to recover Shearer’s optimal bound in terms of the maximum degree Δ(G). We also apply our more sophisticated bounds to estimate the zero-free region of the hard-core model on the triangular lattice (hard hexagons model).

Chain graphs are exactly bipartite graphs without induced 2K 2 (a graph with four vertices and two disjoint edges). A graph G=(V,E) with a given independent set S⊆V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G * is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m 2)-time recognition algorithm for chain partitioned probe graphs.

In this paper, we sketch characteristics of those problems which can be systematically solved on decomposable graphs. Trees, series-parallel graphs, outerplanar graphs, and bandwidth-k graphs all belong to decomposable graphs. Let T d (|V|,|E|) and P d (|V|,|E|) denote the time complexity and processor complexity required to construct a parse tree representation T G for a decomposable G=(V,E) on a PRAM model M d . We define a general problem-solving paradigm to solve a wide class of subgraph optimization problems on decomposable graphs in O(T d (|V|,|E|)+log |V(T G )|) time using O(P d (|V|,|E|)+|V(T G )|/log |V(T G )|) processors on M d . By using our paradigm, we show the following parallel complexities: (a) The maximum independent set problem on trees can be solved in O(log |V|) time using O(|V|/log |V|) processors on an EREW PRAM. (b) The maximum matching problem on series-parallel graphs can be solved in O(log |E|) time using O(|E|/log |E|) processors on an EREW PRAM. (c) The efficient domination problem on series-parallel graphs can be solved in O(log |E|) time using O(|E|/log |E|) processors on an EREW PRAM.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Clustering and domination in perfect graphs

Chordal graphs are those graphs which have chords for each cycle of the length > 3. For large graphs/ networks, generally, the number of chords is less in number than the required number of chords to for a chordal graph. By definition, those graphs are not chordal. The algorithms and properties of chordal graphs do not apply to such cases. Also, the strength of the chords is not measured there. This study introduces a relaxation on such number of chords for the definition of chordal graphs. The notion of a fuzzy strong chord is introduced. After that, fuzzy chordal graphs and related properties are developed. The measure of fuzzy strong chords is proposed. Also, as a generation, fuzzy k-chordal graphs are developed, and isomorphism on a fuzzy chordal graph is defined. At last, area of applications of fuzzy chordal graphs and conclusions with future directions are illustrated.

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.Key wordsSensor network localizationGraph realizationSemidefinite programmingSubject Classifications90C2290C4690C9090B18

In Hell et al. (2004), we have previously observed that, in a chordal graph G, the maximum number of independent r -cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr , with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r -cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r -clique in G ,w eseek a set of independent r -cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r -cliques have the same weight, we simplify the algorithms reported in Hell et al. (2004), although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of $n^{1-epsilon}$, for any $epsilon > 0$. However, there are many ``easy classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an $O(sqrt{n})$-approximation algorithm when the input graph is the union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs) cannot be approximated within any constant factor.