Threshold Graphs Research Articles

The role of the anti-regular graph in the spectral analysis of threshold graphs

The purpose of this paper is to highlight the role played by the anti-regular graph within the class of threshold graphs. Using the fact that every threshold graph contains a maximal anti-regular graph, we show that some known results, and new ones, on the spectral properties of threshold graphs can be deduced from (i) the known results on the eigenvalues of anti-regular graphs, (ii) the subgraph structure of threshold graphs, and (iii) eigenvalue interlacing. In particular, we prove a strengthened version of the recently proved fact that no threshold graph contains an eigenvalue in the interval Ω=[−1−22,−1+22], except possibly the trivial eigenvalues −1 and/or 0, determine the inertia of a threshold graph, and give partial results on a conjecture regarding the optimality of the non-trivial eigenvalues of an anti-regular graph within the class of threshold graphs.

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

AbstractA hypergraph is said to be 1‐Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of ‐Sperner hypergraphs to graphs. First, we consider several ways of associating hypergraphs to graphs, namely, vertex cover, clique, independent set, dominating set, and closed neighborhood hypergraphs. For each of them, we characterize graphs yielding ‐Sperner hypergraphs. These results give new characterizations of threshold and domishold graphs. Second, we apply a characterization of ‐Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems, based on certain matrix partitions, lead to new classes of graphs of bounded clique‐width and new polynomially solvable cases of three basic domination problems: domination, total domination, and connected domination.

Brouwer type conjecture for the eigenvalues of distance signless Laplacian matrix of a graph

Let G be a simple connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues . For , let and be respectively the sum of k-largest distance signless Laplacian eigenvalues and the sum of k-smallest distance signless Laplacian eigenvalues of G. In this paper, we obtain the bounds for and in terms of the number of vertices n and the transmission of the graph G. We propose a Brouwer-type conjecture for and show that it holds for graphs of diameter one and graphs of diameter two for all k. As a consequence, we observe that the conjecture holds for threshold graphs and split graphs (of diameter two). We also show that it holds for k = n−1 and n for all graphs and for some k for r-transmission regular graphs.

On the zero forcing number of a graph involving some classical parameters

Given a simple graph G, let $$Z(G),\, p(G),\, \Phi (G), ex(G)$$ and M(G), respectively, be the zero forcing number, the number of pendant vertices, the cyclomatic number, the number of exterior major vertices and the maximum nullity of G. Wang et al. (Linear Multilinear Algebra, 2018. https://doi.org/10.1080/03081087.2018.1545829) established upper and lower bounds on Z(G) with respect to $$p(G),\, ex(G)$$ and $$\Phi (G)$$: $$p(G)-ex(G)-1\leqslant Z(G)\leqslant p(G)+2\Phi (G)+1$$. Hence, it is interesting to study the distribution of the zero forcing number Z(G) in the interval $$[p(G)-ex(G)-1,\, p(G)+2\Phi (G)+1]$$. Wang et al. (2018) determined all the connected graphs G with $$Z(G)=p(G)-ex(G)$$ and $$Z(G)=p(G)+2\Phi (G)-1.$$ In this paper we identify all the connected graphs G with $$Z(G)=p(G)-ex(G)+1$$ and $$Z(G)=p(G)+2\Phi (G)-2.$$ On the other hand, ‘AIM Minimum Rank-Special Graphs Work Group’ (Linear Algebra Appl 428(7):1628–1648, 2008) established the inequality $$Z(G)\geqslant M(G)$$. The authors posted an attractive question: What is the class of graphs G for which $$Z(G)=M(G)$$? In this paper, we show that the equality holds for threshold graphs.

Algorithmic aspects of upper paired-domination in graphs

A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A paired-dominating set of G is minimal if no proper subset of it is a paired-dominating set of G. The upper paired-domination number of G, denoted by Γpr(G), is the maximum cardinality of a minimal paired-dominating set of G. In Upper-PDS, it is required to compute a minimal paired-dominating set with cardinality Γpr(G) for a given graph G. In this paper, we show that Upper-PDS cannot be approximated within a factor of n1−ε for any ε>0, unless P=NP and Upper-PDS is APX-complete for bipartite graphs of maximum degree 4. On the positive side, we show that Upper-PDS can be approximated within O(Δ)-factor for graphs with maximum degree Δ. We also show that Upper-PDS is solvable in polynomial time for threshold graphs, chain graphs, and proper interval graphs.

Chomp on generalized Kneser graphs and others

In Chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can an explicit strategy be devised? We answer these questions (and determine the Nim-value) for the class of generalized Kneser graphs and for several families of Johnson graphs. We also generalize some of these results to the clique complexes of these graphs. Furthermore, we determine which player has a winning strategy for some classes of threshold graphs.

Note on chromatic polynomials of the threshold graphs

Let G be a threshold graph. In this paper, we give, in first hand, a formula relating the chromatic polynomial of Ḡ (the complement of G ) to the chromatic polynomial of G . In second hand, we express the chromatic polynomials of G and Ḡ in terms of the generalized Bell polynomials.

The complexity of the defensive domination problem in special graph classes

A k-attack on a graph G is a set of k distinct vertices {a1,…,ak} which are said to be under attack. A k-attack A can be countered or defended by a subset of defender vertices X if and only if there exists an injective function f from A to X, such that either f(ai)=ai or (ai,f(ai)) is an edge of G, for all i, 1≤i≤k. Given a graph G, a subset D of V is a k-defensive dominating set of G if and only if D can counter any k-attack in G. We consider the k-defensive dominating set problem.We start a systematic study on the complexity of the k-defensive dominating set problem. When k is not fixed, we show that it is already co-NP-hard to decide whether a given set of vertices is a k-defensive dominating set or not. However, if k is fixed, then we show that the problem is NP-complete even in split graphs. Subsequently, we consider special graph classes, namely paths, cycles, co-chain graphs and threshold graphs. We show that in each of these graph classes, a k-defensive dominating set of minimum size can be found in linear time even for non-fixed k.

On hyper-Hamiltonicity in graphs

A Hamiltonian graph G is said to be hyper-Hamiltonian when G−v is Hamiltonian for any vertex v of G. In this paper, sufficient conditions for a graph to be hyper-Hamiltonian based on different parameters, such as degree and number of edges, are presented. Furthermore, other sufficient conditions for hyper-Hamiltonicity are obtained from spectral parameters, such as the spectral radii of the adjacency matrix, the signless Laplacian matrix and the distance matrix. Results on hyper-Hamiltonicity of a threshold graph through the eigenvalues of its Laplacian matrix are also presented.

The graph tessellation cover number: Chromatic bounds, efficient algorithms and hardness

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number is the size of the smallest tessellation cover. These concepts are motivated by their application to quantum walk models, in special, the evolution operator of the staggered model is obtained from a graph tessellation cover. We show that the minimum between the chromatic index of the graph and the chromatic number of its clique graph, which we call chromatic upper bound, is tight with respect to the tessellation cover number for star-octahedral and windmill graphs; whereas for (3,p)-extended wheel graphs, the tessellation cover number is 3 and the chromatic upper bound is 3p. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. Using graph classes whose tessellation cover numbers achieve the chromatic upper bound, we obtain that t-tessellability is polynomial-time solvable for bipartite, {triangle, proper major}-free, threshold, and diamond-free K-perfect graphs; whereas is NP-complete for triangle-free for t≥3, unichord-free for t≥3, planar for t=3, biplanar for t≥3, chordal (2,1)-graphs for t≥4, (1,2)-graphs for t≥4, and diamond-free with diameter at most five for t=3. We improve the complexity of 2-tessellability problem to linear time.

Eigenvalue-free interval for threshold graphs

This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which −1 and 0 are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval [(−1−2)/2,(−1+2)/2]. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).

The minimum number of Hamilton cycles in a Hamiltonian threshold graph of a prescribed order

AbstractWe prove that the minimum number of Hamilton cycles in a Hamiltonian threshold graph of order is and this minimum number is attained uniquely by the graph with degree sequence of distinct degrees. This graph is also the unique graph of minimum size among all Hamiltonian threshold graphs of order .

Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.

Minimal Laplacian controllability problems of threshold graphs

This study is concerned with the minimal controllability problem of a connected threshold graph following the Laplacian dynamics. The goal is to find the minimum number of controllers and a small set of vertices for the controllers to connect to render the graph Laplacian controllable. A simple algorithm is provided to generate a spanning set of orthogonal Laplacian eigenvectors of the graph from a straightforward computation on its Laplacian matrix. A necessary and sufficient condition for the graph to be Laplacian controllable is then proposed. The condition suggests that the minimum number of controllers to make a connected threshold graph Laplacian controllable is the maximum multiplicity of Laplacian eigenvalues of the graph, and this minimum can be achieved using a binary control matrix. If a controller can be connected to one vertex only, the minimum number is the difference between the number of vertices in the graph and the number of vertices with different degrees. The condition also implies that the controllers ensuring the Laplacian controllability should be connected to the vertices with repeating degrees to break the symmetry of the network topology. Several examples are provided to illustrate the authors' results.

Tropical paths in vertex-colored graphs

A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the initial graph. In this work we study the problem of finding tropical paths in vertex-colored graphs. There are two versions for this problem: the shortest tropical path problem (STPP), i.e., finding a tropical path with the minimum total weight, and the maximum tropical path problem (MTPP), i.e., finding a path with the maximum number of colors possible. We show that both versions of this problems are NP-hard for directed acyclic graphs, cactus graphs and interval graphs. Moreover, we also provide a fixed parameter algorithm for STPP in general graphs and several polynomial-time algorithms for MTPP in specific graphs, including bipartite chain graphs, threshold graphs, trees, block graphs, and proper interval graphs.

On Some closed operations in threshold graphs

This paper is discussing threshold graph. Threshold graph have numerous applications in diverse areas which include computer science and psychology. A threshold graph can be constructed through by repeated application starts with a single vertex, at each step, either a dominating vertex is added, or an isolated vertex is added. The purpose of this paper is to apply some graph operations in a threshold graph and investigating the new graph as result of those operations whether the resulting graph is still a threshold graph or not. We also construct its creation sequence generally.By applying some operations which are given at threshold graph, we can conclude that for union, intersection, ring sum and complement graph, the new graphs are still threshold graphs. Otherwise for the join, the new graphs is not threshold graphs. The creation sequence is achieved generally for each operation.

On the Eigenvalues Distribution in Threshold Graphs

A threshold graph G of order n is defined by binary sequence of length n. In this paper, we consider the adjacent matrix of a connected threshold graph, and give the eigenvalues distribution in threshold graphs. Moreover, we pick out all the threshold graphs with distinct eigenvalues, and determine the HOMO–LUMO index of threshold graphs.

Vertex types in threshold and chain graphs

A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one colour class. Given a graph G, let λ be an eigenvalue (of the adjacency matrix) of G with multiplicity k≥1. A vertex v of G is a downer, or neutral, or Parter depending whether the multiplicity of λ in G−v is k−1, or k, or k+1, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi et al.

On structural parameterizations of firefighting

The Firefighting problem is defined as follows. At time t=0, a fire breaks out at a vertex of a graph. At each time step t≥1, a firefighter permanently defends (protects) an unburned vertex, and the fire then spreads to all undefended neighbors from the vertices on fire. This process stops when the fire cannot spread anymore. The goal is to find a sequence of vertices for the firefighter that maximizes the number of saved (non burned) vertices.The Firefighting problem turns out to be NP-hard even when restricted to bipartite graphs or trees of maximum degree three. We study the parameterized complexity of the Firefighting problem for various structural parameterizations. All our parameters measure the distance to a graph class (in terms of vertex deletion) on which the Firefighting problem admits a polynomial-time algorithm.To begin with, we show that the problem is W[1]-hard when parameterized by the size of a modulator to diameter at most two graphs and split graphs. In contrast to the above intractability results, we show that Firefighting is fixed parameter tractable (FPT) when parameterized by the size of a modulator to cographs, threshold graphs and disjoint unions of stars. We further investigate the kernelization complexity of the problem and show that it does not admit a polynomial kernel when parameterized by the size of a modulator to a disjoint union of stars under some complexity-theoretic assumptions.

Probabilistic and exact frequent subtree mining in graphs beyond forests

Motivated by the impressive predictive power of simple patterns, we consider the problem of mining frequent subtrees in arbitrary graphs. Although the restriction of the pattern language to trees does not resolve the computational complexity of frequent subgraph mining, in a recent work we have shown that it gives rise to an algorithm generating probabilistic frequent subtrees, a random subset of all frequent subtrees, from arbitrary graphs with polynomial delay. It is based on replacing each transaction graph in the input database with a forest formed by a random subset of its spanning trees. This simple technique turned out to be quite powerful on molecule classification tasks. It has, however, the drawback that the number of sampled spanning trees must be bounded by a polynomial of the size of the transaction graphs, resulting in less impressive recall even for slightly more complex structures beyond molecular graphs. To overcome this limitation, in this work we propose an algorithm mining probabilistic frequent subtrees also with polynomial delay, but by replacing each graph with a forest formed by an exponentially large implicit subset of its spanning trees. We demonstrate the superiority of our algorithm over the simple one on threshold graphs used e.g. in spectral clustering. In addition, providing sufficient conditions for the completeness and efficiency of our algorithm, we obtain a positive complexity result on exact frequent subtree mining for a novel, practically and theoretically relevant graph class that is orthogonal to all graph classes defined by some constant bound on monotone graph properties.