Maximum spread of graphs and bipartite graphs

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

Here, we deal with a model of multitype network with nonpreferential attachment growth. The connection between two nodes depends asymmetrically on their types, reflecting the implication of time order in temporal networks. Based upon graph limit theory, we analytically determined the limit of the network model characterized by a kernel, in the sense that the number of copies of any fixed subgraph converges when network size tends to infinity. The results are confirmed by extensive simulations. Our work thus provides a theoretical framework for quantitatively understanding grown temporal complex networks as a whole.

A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γ is (G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5.

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

Clustering and domination in perfect graphs

Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte’s 1-factor theorem and Hall’s matching theorem. All graphs considered in this chapter are loopless.

We present a novel approach for tracking space-filling features, i.e., a set of features covering the entire domain. The assignment between successive time steps is determined by a two-step, global optimization scheme. First, a maximum-weight, maximal matching on a bi-partite graph is computed to provide one-to-one assignments between features of successive time steps. Second, events are detected in a subsequent step; here the matching step serves to restrict the exponentially large set of potential solutions. To this end, we compute an independent set on a graph representing conflicting event explanations. The method is evaluated by tracking dissipation elements, a structure definition from turbulent flow analysis.

If G is a bipartite graph with bipartition A, B then let Gm,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a1, …, am, each vertex b of B by an independent set b1,…, bn, and each edge ab of G by the complete bipartite graph with edges aibj (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of Gm,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess‐current graphs.” They include new genus embeddings for a large class of bipartite graphs.

The generalized vertex cover problem and some variations

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent set. We present properties of different-distance sets, and consider the different-distance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.

A pair of nonnegative integer sequences {D1, D2} with D1 = (d1,1, d1,2, …, dnD2= (d2,1, d2,2, …, dna bipartite graphical sequence, if there is a bipartite graph G with degrees{D1, D2} (i.e., G has two independent vertex sets V1 = {v1,1, v1,2, …, vnV2 = {v2,1, v2,2, …, vnsuch that di,ji the degree of vertex vi,jiG for each i = 1, 2 and ji = 1, 2, …, ni). In other words,{D1, D2} is a bipartite graphical sequence if and only if there is an n1× n2 matrix of 0's and 1's havingdjj1 and djj2. The connectivity κ({D1, D2}) of a bipartite graphical sequence{D1, D2} is defined to be the maximum integer k such that there is a k-connected bipartite graph with degrees {D1, D2}. In this paper, we present a characterization of a k-connected bipartite graphical sequence and an O(n log log n) time algorithm, for a given bipartite graphical sequence {D1, D2}, to construct a κ({D1, D2})-connected bipartite graph with degrees {D1, D2} (n = n1+n2). © 1997 John Wiley & Sons, Inc.

Parity graphs form a superclass of bipartite graphs. Since their introduction, all the algorithms proposed as solutions to the recognition problem and other computational problems exploit the structural property described by Burlet and Uhry in [4].This paper newly describes a different structural property based on split decomposition for parity graphs. This result, together with the observation that the split decomposition process can be performed in linear time, allows us to provide optimum algorithms for both the recognition problem and the maximum weighted clique problem in this class. We further propose a general algorithm to solve the maximum weighted independent set problem for certain classes of graphs. Its application to parity graphs provides the best solution for this problem. Moreover, when the algorithm is applied to distance-hereditary graphs, it equals the best known solution, which is also the optimum for this class.A remarkable consequence of this work is that the extension of bipartite graphs to parity graphs does not increase the complexity of these basic problems, since the worst case occurs when the parity graph is an undecomposable bipartite graph.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.