Special Classes Of Graphs Research Articles

Partitioning subclasses of chordal graphs with few deletions

In the (Vertex)k-Way Cut problem, input is an undirected graph G, an integer s, and the goal is to find a subset S of edges (vertices) of size at most s, such that G−S has at least k connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that k-Way Cut is W[1]-hard parameterized by k. However, Kawarabayashi and Thorup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter s and provided a O(ssO(s)n2) time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time sO(s)nO(1) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertexk-Way Cut is W[1]-hard with respect to either of the parameters, k or s or k+s. These algorithmic results motivate us to look at the problems on special classes of graphs. In this paper, we consider the (Vertex)k-Way Cut problem on subclasses of chordal graphs and obtain the following results.•We first give a sub-exponential FPT algorithm for k-Way Cut running in time 2O(slog⁡s)nO(1) on chordal graphs.•It is “known” that Vertexk-Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k+s. We complement this hardness result by designing polynomial-time algorithms for Vertexk-Way Cut on interval graphs, circular-arc graphs and permutation graphs.

Treewidth versus clique number. II. Tree-independence number

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)-bounded. The family of (tw,ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw,ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent (tw,ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw,ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T. The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw,ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

Stability, Vertex Stability, and Unfrozenness for Special Graph Classes

Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for Θ2P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\Theta _{2}^{\ extrm{P}}}$$\\end{document}, the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\alpha }$$\\end{document} (the independence number), β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\beta }$$\\end{document} (the vertex cover number), ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\omega }$$\\end{document} (the clique number), and χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\chi }$$\\end{document} (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.

Croke–Kleiner Admissible Groups: Property (QT) and Quasiconvexity

Croke–Kleiner admissible groups first introduced by Croke and Kleiner [CK02] belong to a particular class of graphs of groups, which generalizes fundamental groups of three-dimensional graph manifolds. In this paper, we show that if G is a Croke–Kleiner admissible group, then a finitely generated subgroup of G has finite height if and only if it is strongly quasiconvex. We also show that if G↷X is a flip CKA action, then G is quasiisometrically embedded into a finite product of quasitrees. With further assumption on the vertex groups of the flip CKA action G↷X, we show that G satisfies property (QT) introduced by Bestvina, Bromberg, and Fujiwara [BBF21].

The energy complexity of diameter and minimum cut computation in bounded-genus networks

This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only polylog⁡n energy. However, there exist some problems, such as computing the diameter of the graph, that require Ω(n) energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a (1±ϵ)-approximate s-t minimum cut size with O˜(n) energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

On structural parameterizations of load coloring

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring f:V(G)→{r,b} of G such that for every i∈{r,b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time f(w)no(w), unless ETH fails and it can be solved in time nO(w), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is NP-complete even on bipartite graphs and split graphs.

Regular graphs of degree at most four that allow two distinct eigenvalues

For an n×n matrix A, let q(A) be the number of distinct eigenvalues of A. If G is a connected graph on n vertices, let S(G) be the set of all real symmetric n×n matrices A=[aij] such that for i≠j, aij=0 if and only if {i,j} is not an edge of G. Let q(G)=min{q(A):A∈S(G)}. Studying q(G) has become a fundamental sub-problem of the inverse eigenvalue problem for graphs, and characterizing the case for which q(G)=2 has been especially difficult. This paper considers the problem of determining the regular graphs G that satisfy q(G)=2. The resolution is straightforward if the degree of regularity is 1,2, or 3. However, the 4-regular graphs with q(G)=2 are much more difficult to characterize. A connected 4-regular graph has q(G)=2 if and only if either G belongs to a specific infinite class of graphs, or else G is one of fifteen 4-regular graphs whose number of vertices ranges from 5 to 16. This technical result gives rise to several intriguing questions.

Fuzzy Topological Characterization of q C n Graph via Fuzzy Topological Indices

Fuzzy topological indices are one of the accomplished mathematical approaches for numerous technology, engineering, and real-world problems such as telecommunications, social networking, traffic light controls, marine, neural networks, Internet routing, and wireless sensor network (Muneera et al. (2021)). This manuscript comprises the study of a particular class of graphs known as q C n snake graphs. Some innovative results regarding fuzzy topological indices have been established. The major goal of the work is to introduce the notions of First Fuzzy Zagrab Index, Second Fuzzy Zagrab Index, Randic Fuzzy Zagrab Index, and Harmonic Fuzzy Zagrab Index of the q C n snake graph.

Clique Transversal Variants on Graphs: A Parameterized-Complexity Perspective

The clique transversal problem and its variants have garnered significant attention in the last two decades due to their practical applications in communication networks, social-network theory and transceiver placement for cellular telephones. While previous research primarily focused on determining the polynomial-time solvability or NP-hardness/NP-completeness of specific graphs, this paper adopts a parameterized-complexity approach. It thoroughly explores four clique transversal variants: the d-fold transversal problem, the {d}-clique transversal problem, the signed clique transversal problem and the minus clique transversal problem. The paper presents various findings regarding the parameterized complexity of the clique transversal problem and its variants. It establishes the W[2]-completeness and para-NP-completeness of the d-fold transversal problem, the {d}-clique transversal problem, the signed clique transversal problem and the minus clique transversal problem within specific graph classes. Additionally, it introduces fixed-parameter tractable algorithms for planar graphs and graphs with bounded treewidth, offering efficient solutions for these specific instances of the problems. The research further explores the relationship between planar graphs and graphs with bounded treewidth to enhance the time complexity of the d-fold clique transversal problem and the {d}-clique transversal problem. By analyzing the parameterized complexity of the clique transversal problem and its variants, this research contributes to our understanding of the computational limitations and potentially efficient algorithms for solving these problems.

On D-distance and D-closed Graphs

This paper studies the concepts of D-boundary vertex, D-interior vertex, D-null vertex, D-closure of a graph and D-closed graph, based on the idea of D-distance. We investigate the structural properties of these concepts and determine whether some special classes of graphs are D-closed or not.

Deletion to scattered graph classes II - improved FPT algorithms for deletion to pairs of graph classes

The problem of deletion of vertices to a hereditary graph class is a well-studied problem in parameterized complexity. Recently, a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes and we determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be fixed parameter tractable (FPT) when the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. This paper focuses on pairs of specific graph classes (Π1,Π2) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms.

Some upper and lower bounds for $D_{\alpha}$-energy of graphs

The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$. Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalizeddistance energy of $G$ can be defined as $E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$. In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for$E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.

Some Properties of Chain and Threshold Graphs

Chain graphs and threshold graphs are special classes of graphs which have maximum spectral radius among bipartite graphs and connected graphs with given order and size, respectively. In this article, we focus on some of linear algebraic tools like rank, determinant, and permanent related to the adjacency matrix of these types of graphs. We derive results relating the rank and number of edges. We also characterize chain/threshold graphs with nonzero determinant and permanent.

On Non-Zero Vertex Signed Domination

For a graph G=(V,E) and a function f:V→{−1,+1}, if S⊆V then we write f(S)=∑v∈Sf(v). A function f is said to be a non-zero vertex signed dominating function (for short, NVSDF) of G if f(N[v])=0 holds for every vertex v in G, and the non-zero vertex signed domination number of G is defined as γsb(G)=max{f(V)|f is an NVSDF of G}. In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of γsb(G) for several special classes of graphs is determined. Finally, we pose some open problems.

Cosmological states in loop quantum gravity on homogeneous graphs

We introduce a class of states characterized by proposed conditions of homogeneity and isotropy in loop quantum gravity and construct concrete examples given by Bell-network states on a special class of homogeneous graphs. Such states provide new representations of cosmological spaces that can be explored for the formulation of cosmological models in the context of loop quantum gravity. We show that their local geometry is described in an automorphism-invariant manner by one-node observables analogous to the one-body observables used in many-body quantum mechanics, and compute the density matrix representing the restriction of global states to the algebra of one-node observables. The von Neumann entropy of this density matrix provides a notion of entanglement entropy of a local region which respects automorphism-invariance and can be applied to states involving superpositions of distinct graphs.

Better bounds on the adaptivity gap of influence maximization under full-adoption feedback

In the influence maximization (IM) problem, we are given a social network and a budget k, and we look for a set of k nodes in the network, called seeds, that maximize the expected number of nodes that are reached by an influence cascade generated by the seeds, according to some stochastic model for influence diffusion. Extensive studies have been done on the IM problem, since this definition by Kempe et al. [26]. However, most of the work focuses on the non-adaptive version of the problem where all the k seed nodes must be selected before the cascade starts. In this paper we study the adaptive IM, where the nodes are selected sequentially one by one, and the decision on the i-th seed can be based on the observed cascade produced by the first i−1 seeds. We focus on the full-adoption feedback in which we can observe the entire cascade of each previously selected seed under the independent cascade model where each edge is associated with an independent probability of diffusing influence.Previous works showed that there are constant upper bounds on the adaptivity gap, which compares the performance of an adaptive algorithm against a non-adaptive one, but the analyses used to prove these bounds only work for specific graph classes such as in-arborescences, out-arborescences, and one-directional bipartite graphs. Our main result is the first sub-linear upper bound that holds for any graph. Specifically, we show that the adaptivity gap is upper-bounded by n3+1, where n is the number of nodes in the graph. Moreover, we improve over the known upper bound for in-arborescences from 2e/(e−1)≈3.16 to 2e2/(e2−1)≈2.31. Then, we consider (β,γ)-bounded-activation graphs, where all nodes but β influence in expectation at most γ∈[0,1) neighbors each; for this class of influence graphs we show that the adaptivity gap is at most β+11−γ. Finally, we study α-bounded-degree graphs, that is the class of undirected graphs in which the sum of node degrees higher than two is at most α, and show that the adaptivity gap is upper-bounded by α+O(1); we also show that in 0-bounded-degree graphs, i.e. undirected graphs in which each connected component is a path or a cycle, the adaptivity gap is at most 3e3/(e3−1)≈3.16.To prove our bounds, we introduce new techniques to relate adaptive policies with non-adaptive ones that might be of their own interest.

On finding separators in temporal split and permutation graphs

We study the NP-hard Temporal(s,z)-Separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle Temporal(s,z)-Separation by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general Temporal(s,z)-Separation remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of “change over time”.

The complexity landscape of claim-augmented argumentation frameworks

Claim-augmented argumentation frameworks (CAFs) provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective; they extend Dung AFs by associating a claim to each argument representing its conclusion. This additional layer offers various possibilities to generalize abstract argumentation semantics, i.e. the re-interpretation of arguments in terms of their claims can be performed at different stages in the evaluation of the framework: One approach is to perform the evaluation entirely at argument-level before interpreting arguments by their claims (inherited semantics); alternatively, one can perform certain steps in the process (e.g., maximization) already in terms of the arguments' claims (claim-level semantics). The inherent difference of these approaches not only potentially results in different outcomes but, as we will show in this paper, is also mirrored in terms of computational complexity. To this end, we provide a comprehensive complexity analysis of the four main reasoning problems with respect to claim-level variants of preferred, naive, stable, semi-stable and stage semantics and complete the complexity results of inherited semantics by providing corresponding results for semi-stable and stage semantics. Furthermore, we provide complexity results for these types of frameworks when restricted to specific graph classes and when parameterized by the number of claims within the framework. Moreover, we show that deciding, whether for a given framework the two approaches of a semantics coincide (concurrence) can be surprisingly hard, ranging up to the third level of the polynomial hierarchy.

The complexity of bicriteria tree-depth

The tree-depth problem can be seen as finding an elimination tree of minimum height for a given input graph G. We introduce a bicriteria generalization in which additionally the width of the elimination tree needs to be bounded by some input integer b. We are interested in the case when G is the line graph of a tree, proving that the problem is NP-hard and obtaining a polynomial-time additive 2b-approximation algorithm. This particular class of graphs received significant attention in the past, mainly due to a number of potential applications, e.g. in parallel assembly of modular products, or parallel query processing in relational databases, as well as purely combinatorial applications including searching in tree-like partial orders (which in turn generalizes binary search on sorted data).

Mutual-visibility in distance-hereditary graphs: a linear-time algorithm

The concept of mutual-visibility in graphs has been recently introduced. If X is a subset of vertices of a graph G, then vertices u and v are X-visible if there exists a shortest u, v-path P such that V(P) ∩ X⊆ {u, v}. If every two vertices from X are X-visible, then X is a mutual-visibility set. The mutual-visibility number of G is the cardinality of a largest mutual-visibility set of G. It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class.