Chordal Research Articles

MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

Algebraic implications of neighborhood hypergraphs and their transversal hypergraphs

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals.

Sharp Lower Bound of Cacti Graph with respect to Zagreb Eccentricity Indices

The first Zagreb eccentricity index E1(℧) is the sum of square of eccentricities of the vertices, and the second Zagreb eccentricity index E2(℧) is the sum of product squares of the eccentricities of the vertices. A linked graph G is called a cactus if any two of its cycles share only one vertex. In other words, there are no two independent cycles that share an edge. Cactus graphs are also known as “block graphs” or “sensitized graphs.” They are closely related to chordal graphs and can be used to represent various types of networks, including communication networks and road networks. In this contribution, E1(℧) and E2(℧) values of cacti with k pendant vertices and k cycles, respectively, are considered. We determine the minimum E1, E2 indices for n order cacti with k pendant vertices and k cycles.

H - V -Super-Strong- ( a , d ) -antimagic decomposition of complete bipartite graphs

An H -(a,d)-antimagic labeling in a H -decomposable graph G is a bijection f : V ( G ) ∪ E ( G ) → { 1 , 2 , … , p + q } such that ∑ f ( H 1 ) , ∑ f ( H 2 ) , ⋯ , ∑ f ( H h ) forms an arithmetic progression with difference d and first element a . f is said to be H - V -super- ( a , d ) -antimagic if f ( V ( G ) ) = { 1 , 2 , … , p } . Suppose that V ( G ) = U ( G ) ∪ W ( G ) with | U ( G ) | = m and | W ( G ) | = n . Then f is said to be H - V -super-strong- ( a , d ) -antimagic labeling if f ( U ( G ) ) = { 1 , 2 , … , m } and f ( W ( G ) ) = { m + 1 , m + 2 , … , ( m + n = p ) } . A graph that admits a H - V -super-strong- ( a , d ) -antimagic labeling is called a H - V -super-strong- ( a , d ) -antimagic decomposable graph. In this paper, we prove that complete bipartite graphs K m , n are H - V -super-strong- ( a , d ) -antimagic decomposable with both m and n are even.

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

The leafage of a chordal graph G is the minimum integer ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} such that G can be realized as an intersection graph of subtrees of a tree with ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2O(ℓ2)·nO(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{\\mathcal {O}(\\ell ^2)} \\cdot n^{\\mathcal {O}(1)}$$\\end{document}. We present a conceptually much simpler algorithm that runs in time 2O(ℓ)·nO(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{\\mathcal {O}(\\ell )} \\cdot n^{\\mathcal {O}(1)}$$\\end{document}. We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple nO(ℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{\\mathcal {O}(\\ell )}$$\\end{document}-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in nO(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{{{\\mathcal {O}}}(1)}$$\\end{document}-time.

Roman [1,2]-domination of graphs

A [1,2]-set of a graph G is a set S of vertices of G if every vertex not in S has one or two neighbors in S. The [1,2]-domination number γ[1,2](G) of G equals the minimum cardinality of a [1,2]-set of G. A Roman [1,2]-dominating function (R[1,2]DF) on a graph G is a function f from the vertex set V of G to the set {0,1,2} such that any vertex assigned 0 under f has one or two neighbors assigned 2. The weight of an R[1,2]DF f is the sum ∑x∈Vf(x). The Roman [1,2]-domination number γR[1,2](G) of G equals the minimum weight of an R[1,2]DF on G. In this paper, we prove that the decision problem on the Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. Moreover, we give some bounds on the Roman [1,2]-domination number. In particular, we show that for any nontrivial tree T, γR[1,2](T)≥γ[1,2](T)+1 and characterize all trees obtaining equality in this bound.

S-Club Cluster Vertex Deletion on interval and well-partitioned chordal graphs

In this paper, we study the computational complexity of s-Club Cluster Vertex Deletion. Given a graph, s-Club Cluster Vertex Deletion (s-CVD) aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most s. When s=1, the corresponding problem is popularly known as Cluster Vertex Deletion (CVD). We provide a faster algorithm for s-CVD on interval graphs. For each s≥1, we give an O(n(n+m))-time algorithm for s-CVD on interval graphs with n vertices and m edges. In the case of s=1, our algorithm is a slight improvement over the O(n3)-time algorithm of Cao et al. (2018), and for s≥2, it significantly improves the state-of-the-art running time On4.We also give a polynomial-time algorithm to solve CVD on well-partitioned chordal graphs, a graph class introduced by Ahn et al. (WG 2020) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the Weighted Bipartite Vertex Cover. This generalises a result of Cao et al. (2018) on split graphs. We also show that for any even integer s≥2, s-CVD is NP-hard on well-partitioned chordal graphs.

Solution to a problem of Katona on counting cliques of weighted graphs

A subset I of the vertex set V(G) of a graph G is called a k-clique independent set ofG if no k vertices in I form a k-clique of G. An independent set is a 2-clique independent set. Let πk(G) denote the number of k-cliques of G. For a function w:V(G)→{0,1,2,…}, let G(w) be the graph obtained from G by replacing each vertex v by a w(v)-clique Kv and making each vertex of Ku adjacent to each vertex of Kv for each edge {u,v} of G. For an integer m≥1, consider any w with ∑v∈V(G)w(v)=m. For U⊆V(G), we say that w is uniform onU if w(v)=0 for each v∈V(G)∖U and, for each u∈U, w(u)=m/|U| or w(u)=m/|U|. Katona asked if πk(G(w)) is smallest when w is uniform on a largest k-clique independent set of G. He placed particular emphasis on the Sperner graph Bn, given by V(Bn)={X:X⊆{1,…,n}} and E(Bn)={{X,Y}:X⊊Y∈V(Bn)}. He provided an affirmative answer for k=2 (and any G). We determine graphs for which the answer is negative for every k≥3. These include Bn for n≥2. Generalizing Sperner’s Theorem and a recent result of Qian, Engel and Xu, we show that πk(Bn(w)) is smallest when w is uniform on a largest independent set of Bn. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

Succinct data structure for path graphs

We consider the problem of designing a succinct data structure for path graphs, that generalizes interval graphs, on n vertices while efficiently supporting degree, adjacency, and neighbourhood queries. We provide the following two solutions for this problem:1.an nlog⁡n+o(nlog⁡n)-bit succinct data structure that supports adjacency query in O(log⁡n) time, neighbourhood query in O(dlog⁡n) time and finally, degree query in min⁡{O(log2⁡n),O(dlog⁡n)} time where d is the degree of the queried vertex.2.an O(nlog2⁡n)-bit space-efficient data structure that supports adjacency, neighborhood, and degree queries optimally. Central to our data structures is the usage of the heavy path decomposition, followed by careful bookkeeping using an orthogonal range search data structure using wavelet trees among others, which may be of independent interest for designing succinct data structures for other graph classes.

Strictly chordal graphs: Structural properties and integer Laplacian eigenvalues

In this paper, we establish the relation between classic invariants and integer Laplacian eigenvalues of strictly chordal graphs, pointing out how their computation can be efficiently implemented. We review results concerning general graphs showing that the number of universal vertices and the degrees of twins provide integer Laplacian eigenvalues and their respective multiplicities. For strictly chordal graphs, we show new results about integer Laplacian eigenvalues which are directly related to particular vertex sets of the graph.

Dirac's theorem on chordal graphs implies Brooks' theorem

We give yet another proof of the list-color version of Brooks' theorem that is due, independently, to Vizing and to Erdős, Rubin and Taylor, via a famous theorem of Dirac on chordal graphs.

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].

Prime Graph Generation through Single Edge Addition: Characterizing a Class of Graphs

A graph G consists of a finite set V (G) of vertices with a collection E(G) of unordered pairs of distinct vertices called edge set of G. Let G be a graph. A set M of vertices is a module of G if, for vertices x and y in M and each vertex z outside M, {z, x} ∈ E(G) ⇐⇒ {z, y} ∈ E(G). Thus, a module of G is a set M of vertices indistinguishable by the vertices outside M. The empty set, the singleton sets and the full set of vertices represent the trivial modules. A graph is indecomposable if all its modules are trivial, otherwise it is decomposable. Indecomposable graphs with at least four vertices are prime graphs. The introduction and the study of the construction of prime graphs obtained from a given decomposable graph by adding one edge constitue the central points of this paper.

A polynomial time algorithm for the triangle packing problem on interval graphs

The triangle packing problem (TPP) is to find the maximum number of pairwise vertex disjoint triangles in a given graph. The TPP is NP-complete in a general graph and even so when a given graph is restricted to a chordal graph. On the other hand, the TPP can be solved in polynomial time for unit interval graphs. For this reason, it is a well-known open problem to prove whether there exists a polynomial time algorithm solving the TPP on interval graphs. In this paper, we give an answer to the problem.

Chordal matroids arising from generalized parallel connections

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple GF(q)-representable matroids that can be built from projective geometries over GF(q) by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors. We characterize the class by its forbidden induced minors; the case when q=2 is distinctive.

Algorithms and hardness results for edge total domination problem in graphs

For a graph G=(V,E) without an isolated edge, a set D⊆E is an edge total dominating set of G if every edge e∈E is adjacent to at least one edge of D. The Minimum Edge Total Dominating Set problem is to compute a minimum cardinality edge total dominating set of G. It is known that the decision version of the problem is NP-complete for bipartite graphs, chordal graphs, and planar graphs. On the positive side, the problem is efficiently solvable for trees. In this paper, we further study the complexity of this problem in other graph classes. We design a linear-time algorithm to compute a minimum cardinality edge total dominating set in chain graphs, a subclass of bipartite graphs. We also propose linear-time algorithms for two subclasses of chordal graphs, namely, split graphs and proper interval graphs with no cut vertices. In addition, we show that the problem is APX-complete for graphs with maximum degree 3 and propose an approximation algorithm for the problem in k-regular graphs, where k≥4. We discuss the complexity difference between the Minimum Edge Dominating Set problem and Minimum Edge Total Dominating Set problem which seem to be closely related.

Algorithmic Aspects of Total Vertex-Edge Domination in Graphs

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

Powers of vertex cover ideals of simplicial trees

AbstractIn 2011, Herzog et al. conjectured that if is the cover ideal of a chordal graph, then is componentwise linear for all . In 2022, Hà and Van Tuyl considered objects more general than chordal graphs and posed the following problem: let be the cover ideal of a simplicial tree . Is it true that is componentwise linear for all In this paper, we give an affirmative answer to this problem.