Simplicial Vertices Research Articles

Improving the Caro–Wei bound and applications to Turán stability

We prove that if G is a graph and f(v)≤1/(d(v)+1/2) for each v∈V(G), then either G has an independent set of size at least ∑v∈V(G)f(v) or G contains a clique K such that ∑v∈Kf(v)>1. This result implies that for any σ≤1/2, if G is a graph and every clique K⊆V(G) has at most (1−σ)(|K|−σ) simplicial vertices, then α(G)≥∑v∈V(G)1/(d(v)+1−σ). Letting σ=0 implies the famous Caro–Wei Theorem, and letting σ=1/2 implies that if fewer than half of the vertices in each clique of G are simplicial, then α(G)≥∑v∈V(G)1/(d(v)+1/2), which is tight for the 5-cycle. When applied to the complement of a graph, this result implies the following new Turán stability result. If G is a Kr+1-free graph with more than (1−1/r)n2/2−n/4 edges, then G contains an independent set I such that at least half of the vertices in I are complete to G−I. Applying this stability result iteratively provides a new proof of the stability version of Turán’s Theorem in which Kr+1-free graphs with close to the extremal number of edges are r-partite.

Variety of General Position Problems in Graphs

Let X be a vertex subset of a graph G. Then u,v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u, v\\in V(G)$$\\end{document} are X-positionable if V(P)∩X⊆{u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(P)\\cap X \\subseteq \\{u,v\\}$$\\end{document} holds for any shortest u, v-path P. If each two vertices from X are X-positionable, then X is a general position set. The general position number of G is the cardinality of a largest general position set of G and has been already well investigated. In this paper a variety of general position problems is introduced based on which natural pairs of vertices are required to be X-positionable. This yields the total (resp. dual, outer) general position number. It is proved that the total general position sets coincide with sets of simplicial vertices, and that the outer general position sets coincide with sets of mutually maximally distant vertices. It is shown that a general position set is a dual general position set if and only if its complement is convex. Several sufficient conditions are presented that guarantee that a given graph has no dual general position set. The total general position number, the outer general position number, and the dual general position number of arbitrary Cartesian products are determined.

Bisimplicial separators

AbstractA 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.

Componentwise linearity of powers of cover ideals

Let G be a finite simple graph and J(G) denote its vertex cover ideal in a polynomial ring over a field. The k-th symbolic power of J(G) is denoted by \(J(G)^{(k)}\). In this paper, we give a criterion for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that \(J(G)^{(k)}\) is a componentwise linear ideal for some (equivalently, for all) \(k \ge 2\) when G is a graph such that \(G {\setminus } N_G[A]\) has a simplicial vertex for any independent set A of G. Using this result, we prove that \(J(G)^{(k)}\) is a componentwise linear ideal for several classes of graphs for all \(k \ge 2\). In particular, if G is a bipartite graph, then J(G) is a componentwise linear ideal if and only if \(J(G)^k\) is a componentwise linear ideal for some (equivalently, for all) \(k \ge 2\).

Graphs with a unique maximum independent set up to automorphisms

If for every two maximum independent sets A and B in a graph G there exists an automorphism of G that maps A to B , then we call G an α -iso-unique graph. We extend several known characterizations of trees that have a unique maximum independent set obtaining characterizations of α -iso-unique trees. Such trees either have the property that the deletion of any edge does not affect the independence number, or they have the central edge, which connects two isomorphic subtrees, and only the deletion of the central edge increases the independence number. One of the characterizations results in a linear time algorithm to recognize whether a given tree is α -iso-unique. In contrast, we prove that the problem of recognizing whether an arbitrary graph is α -iso-unique is not polynomial unless P = NP. Constructions of large families of α -iso-unique graphs are given involving simplicial vertices and chordal graphs. In particular, we show that every graph is an induced subgraph of an α -iso-unique graph. Finally, α -iso-unique graphs are characterized among Cartesian products G □ H of co-prime graphs G and H such that α ( G □ H ) = α ( H ) | V ( G ) | .

Topological indices of maximal outerplane graphswith two simplicial vertices

Topological indices of maximal outerplane graphswith two simplicial vertices

Avoidable vertices and edges in graphs: Existence, characterization, and applications

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF-vertices), simplicial paths, and their common generalization avoidable paths. In an earlier version of this paper, published as an extended abstract in the proceedings of the 16th International Symposium on Algorithms and Data Structures (WADS 2019), we presented a general conjecture on the existence of avoidable paths. The conjecture was recently proved by Bonamy et al. (2020). The conjecture—now a theorem—implies a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices and a result due to Chvátal, Sritharan, and Rusu from 2002 on the existence of simplicial paths in graphs excluding all sufficiently long induced cycles. In turn, both of these results generalize Dirac’s classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with at least two edges has two avoidable edges, which strengthens the statement of the theorem of Bonamy et al. for the case of paths of length one. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in two classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the results about the existence of avoidable vertices and edges have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

On the structure and clique‐width of (4K1,C4,C6,C7)‐free graphs

AbstractWe give a complete structural description of ‐free graphs that do not contain a simplicial vertex, and we prove that such graphs have bounded clique‐width. Together with the results of Foley et al., this implies that ‐free graphs that do not contain a simplicial vertex have bounded clique‐width. Consequently, Graph Coloring can be solved in polynomial time for ‐free graphs, that is, for even‐hole‐free graphs of stability number at most three.

An improved exact algorithm for minimum dominating set in chordal graphs

Minimum Dominating Set is among the most studied classical NP-hard problems. On general graphs, an O(1.4864n) exact algorithm for this problem is known. Minimum Dominating Set remains NP-hard on chordal graphs where the current asymptotically-fastest exact algorithm solves the problem in O(1.3687n) time and super-polynomial space. In this paper, a simple exact polynomial-space algorithm that exploits the local structure of neighborhoods of simplicial vertices in chordal graphs is presented, resulting in an improved running time in O(1.3384n).

The open detour number of a graph

A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text].

Graphs in which all maximal bipartite subgraphs have the same order

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered.

Clustering the biological networks using shortest path.

Abstract The aim of graph clustering is to partition the given graph into subgraphs, and then assign the subgraphs to overlapping or non-overlapping clusters. In this process, densely connected subgraphs fall into a single cluster, thereby maximizing the intra-cluster similarity. Here we propose a clustering method based on the shortest path between the nodes. We propose a 2- partition in a few graph classes namely outerplanar graphs, graphs with simplicial vertices, chordal graphs, distance hereditary graphs and cographs. The solution to the chordal and outerplanar graphs can be found in linear time. Chordal graphs are widely used in perfect phylogeny related research whereas cographs are used to model the orthology relations. Hence a clustering method based on shortest path in chordal graphs and cographs may have significant impact in the research related to orthologous genes as well as phylogenetic relations.

Boundary-type sets in maximal outerplanar graphs

The periphery Per(G) of a graph G is the set of vertices of maximum eccentricity. A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than the eccentricity of v. The extreme set Ext(G) of G is the set of all its simplicial (also called extreme) vertices; an extreme vertex is such that its neighborhood induces a complete graph. The eccentricity Ecc(G) of a graph G is the set of all its eccentric vertices, i.e. vertices that are antipodal to some other vertex in G. A vertex v is a boundary vertex if there is another vertex u in G such that no u-v geodesic can be extended at v to a longer geodesic. The boundary∂(G) of G is the set of all its boundary vertices.For the family of maximal outerplanar graphs, we provide a characterization of ∂(G) and Ext(G) in terms of vertex degrees. We characterize those graphs that are induced by Per(G) and Ct(G). We show that, unlike for trees, all relationships between boundary-type sets in maximal outerplanar graphs are as rich as in general graphs and we construct a single maximal outerplanar graph showing the sharpness of all those inclusions.

Brion’s theorem for Gelfand–Tsetlin polytopes

This work is motivated by the observation that the character of an irreducible gln-module (a Schur polynomial), being the sum of exponentials of integer points in a Gelfand–Tsetlin polytope, can be expressed by using Brion’s theorem. The main result is that, in the case of a regular highest weight, the contributions of all nonsimplicial vertices vanish, while the number of simplicial vertices is n! and the contributions of these vertices are precisely the summands in Weyl’s character formula.

End simplicial vertices in path graphs

End simplicial vertices in path graphs

Excluded vertex-minors for graphs of linear rank-width at most [formula omitted

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite obstruction set Ok of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in Ok. However, no attempts have been made to bound the number of graphs in Ok for k≥2. We show that for each k, there are at least 2Ω(3k) pairwise locally non-equivalent graphs in Ok, and therefore the number of graphs in Ok is at least double exponential.To prove this theorem, it is necessary to characterize when two graphs in Ok are locally equivalent. A graph is a block graph if all of its blocks are complete graphs. We prove that if two block graphs without simplicial vertices of degree at least 2 are locally equivalent, then they are isomorphic. This not only is useful for our theorem but also implies a theorem of Bouchet (1988) stating that if two trees are locally equivalent, then they are isomorphic.

A Dirac-type characterization of [formula omitted]-chordal graphs

A graph is k-chordal if it has no induced cycle of length greater than k. We give a new characterization of k-chordal graphs that generalizes the well-known characterization of chordal graphs by Dirac in terms of simplicial vertices and simplicial orderings.

Simplicial Vertices in Graphs with no Induced Four-Edge Path or Four-Edge Antipath, and theH6-Conjecture

Let be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos [6] conjectured that every prime graph in not isomorphic to the cycle of length five is either a split graph or contains a certain useful arrangement of simplicial and antisimplicial vertices. In this article, we give a counterexample to their conjecture, and prove a slightly weaker version. Additionally, applying a result of the first author and Seymour [1] we give a short proof of Fouquet's result [3] on the structure of the subclass of bull-free graphs contained in .

On the hull number of some graph classes

In this paper, we study the geodetic convexity of graphs, focusing on the problem of the complexity of computing a minimum hull set of a graph in several graph classes.For any two vertices u,v∈V of a connected graph G=(V,E), the closed interval I[u,v] of u and v is the set of vertices that belong to some shortest (u,v)-path. For any S⊆V, let I[S]=⋃u,v∈SI[u,v]. A subset S⊆V is geodesically convex or convex if I[S]=S. In other words, a subset S is convex if, for any u,v∈S and for any shortest (u,v)-path P, V(P)⊆S. Given a subset S⊆V, the convex hull Ih[S] of S is the smallest convex set that contains S. We say that S is a hull set of G if Ih[S]=V. The size of a minimum hull set of G is the hull number of G, denoted by hn(G). The Hull Number problem is to decide whether hn(G)≤k, for a given graph G and an integer k. Dourado et al. showed that this problem is NP-complete in general graphs.In this paper, we answer an open question of Dourado et al. (2009) [1] by showing that the Hull Number problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial-time algorithms to solve the Hull Number problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in Araujo et al. (2011) [2] to the class of (q,q−4)-graphs and to cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an n-node graph G without simplicial vertices is at most 1+⌈3(n−1)5⌉ in general, at most 1+⌈n−12⌉ if G is regular or has no triangle, and at most 1+⌈n−13⌉ if G has girth at least 6.

Special Eccentric Vertices for the Class of Chordal Graphs and Related Classes

A vertex is simplicial if the vertices of its neighborhood are pairwise adjacent. It is known that, for every vertex v of a chordal graph, there exists a simplicial vertex among the vertices at maximum distance from v. Here we prove similar properties in other classes of graphs related to that of chordal graphs. Those properties will not be in terms of simplicial vertices, but in terms of other types of vertices that are used to characterize those classes.