Vertex cut of a graph and connectivity of its neighbourhood complex
Rekha Santhanam, Samir Shukla
备注：
Accepted journal version. Incorporated referee's comments. In section 6, an example of a class of graphs is added, where the connectivity of its neighborhood complex can be made arbitrary large from its vertex connectivity
We show that if a graph $G$ satisfies certain conditions then the connectivity of neighbourhood complex $\mathcal{N}(G)$ is strictly less than the vertex connectivity of $G$. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Further, we prove that for a graph $G$ if there exists a vertex $v$ satisfying the property that for any $k$-subset $S$ of neighbours of $v$, there exists a vertex $v_S \neq v$ such that $S$ is subset of neighbours of $v_S$, then $\mathcal{N}(G-\{v\})$ is $(k-1)$-connected implies that $\mathcal{N}(G)$ is $(k-1)$-connected. As a consequence of this, we show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if $G$ is a $(n+1)$-connected chordal graph which is not folded onto a clique of size $n+2$, then $\mathcal{N}(G)$ is $n$-connected.
We derive a new upper bound on the algebraic connectivity of a regular graph using the Higman-Sims technique. Together with a new result on the connectivity of the neighbourhood graph of strongly regular graphs, our result gives a characterization of a class of strongly regular graphs that maximize the algebraic connectivity amongst regular graphs.
Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fàbrega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a \emph{cutset} if $G-S$ is not connected. A cutset $S$ is called an \emph{$R_g$-cutset}, where $g$ is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_g$-cutset, the \emph{$g$-extra connectivity} of $G$, denoted by $κ_g(G)$, is then defined as the minimum cardinality over all $R_g$-cutsets of $G$. In this paper…
In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts re…
A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there is no isolated vertex in each component of $G-S$. The super-connectivity of graph $G$ is the size of the minimum super vertex cut of $G$. Let $KG(n,k)$ be the Kneser graph whose vertices set are the $k$-subsets of $\{1,\cdots,n\}$, where $k$ is the number of labels of each vertex in $G$. We aim to show that the conjecture from Boruzanli and Gauci \cite{EG19} …
A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We first establish that, for any $d\ge 3$, for any $d\ge 3$, every minimum edge cut of cardinality at most $4d-7$ in such a graph is \textit{trivial}, namely it consists of all the edges incident with some vertex. A consequence of this is that, for $d\ge 3$, the graph of a simplicial $d$-polytope with minimum degree $δ$ is $\min\{δ,4d-6\}$-edge-connected. In the particular case of $d=3$, we have that every minimum edge …
For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected graph without isolated vertices. This paper provides bounds for the super connectivity and $k_{1}$-connectivity of the lexicographic product of two graphs.
Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $λ(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $λ(G)=δ(G)$, and $G$ super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edge-connectivity $λ'(G)$ of $G$ is the minimum number of edges whose removal disconnects $G$ into non-trivial components. If $λ'(G)$ achieves the upper bound of restricted edge-connectivity, then $G$ is said to be $λ'$-optimal. A bipartite multigraph is said to be half-t…
A digraph X=(V, E) is max-λ, if λ(X)=δ(X). A digraph X is super-λif every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all but a few exceptions, the strongly connected mixed Cayley digraphs are max-λand super-λ.
There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is $Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edg…
The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$. The edge connectivity of a graph $G$, denoted as $λ(G)$, is the size of a minimum edge-cut in $G$. We introduce a function $ψ$ and prove the following formula %for the edge-connectivity of direct products $$λ(G\times H)=\min {2λ(G)|E(H)|,2λ(H)|E(G)|,δ(G\times H), ψ(G,H), ψ(H,G)} .$$ We also describe the structure of every minimum edge…
Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc. The \emph{restricted arc-connectivity} $λ'(D)$ of a digraph $D$ is the minimum cardinality over all restricted arc-cuts of $D$. A strongly connected digraph $D$ is \emph{$λ'$-connected} when $λ'(D)$ exists. This paper presents a family $\cal{F}$ of strong digraphs of girth four that are not $λ'$-connected and for every strong digraph $D\notin \cal{F}$ with girth four it follows that it is $λ'$-connected.…
A graph is said to be cyclic $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of $k$ edges is called a cyclic-$k$-edge cutset and it is called a trivial cyclic-$k$-edge cutset if at least one of the resulting two components induces a single $k$-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene $F$ containing a nontrivial cyclic-5-edge cutset admits two antipodal pent…
The $g$-extra connectivity is an important parameter to measure the ability of tolerance and reliability of interconnection networks. Given a connected graph $G=(V,E)$ and a non-negative integer $g$, a subset $S\subseteq V$ is called a $g$-extra cut of $G$ if $G-S$ is disconnected and every component of $G-S$ has at least $g+1$ vertices. The cardinality of the minimum $g$-extra cut is defined as the $g$-extra connectivity of $G$, denoted by $κ_g(G)$. In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph…
Let P be a graph property. A graph is locally P if the subgraph induced by the open neighbourhood of every vertex has property P. A graph has the Dirac condition if the minimum degree of every vertex is at least half the order of the graph and it satisfies the Ore condition if the sum of the degrees of any pair of non-adjacent vertices is at least the order of the graph. In this paper we study global properties of graphs that possess the local Dirac and Ore conditions. We focus on the connectivity, edge-connectivity, diameter, planarity and cycle structure of these graphs.
The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity.
The structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus is studied. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained.
Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity $4$ from 1988.
Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, the edges may be weighted with weight set $W =\{w_1, w_2, ..., w_m\}$, the label cut problem(hedge connectivity) problem asks for the minimum number of edge sets(each edge set (or hedge) is the edges with the same label) whose removal disconnects the source-sink pair of vertices or the whole graph with minimum total weights(minimum cardi…
The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal $v$-$w$ entry is the weight of a minimum edge cut separating vertices $v$ and $w$. Its computation is a classical topic of combinatorial optimization since at least the seminal work of Gomory and Hu. In this article, we investigate spectral properties of these matrices. In particular, we provide tight bounds on the smallest eigenvalue and the energy. Moreover, we study the eigenvector structure and show in which cases eigenvectors can be easily obtained from matrix entries. These results in turn rely on a new cha…
Let $κ'(G)$ be the edge connectivity of $G$ and $G\times H$ the direct product of $G$ and $H$. Let $H$ be an arbitrary dense graph with minimal degree $δ(H)>|H|/2$. We prove that for any graph $G$, $κ'(G\times H)=\textup{min}\{2κ'(G)e(H),δ(G)δ(H)\}$, where $e(H)$ denotes the number of edges in $H$. In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for $G\times K_n(n\ge3)$ to be super edge connected.
The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime $3^{+2}$…
Maximum spectral gap of regular graphs with bounded essential edge-connectivity
College of Mathematics and System Science, Xinjiang University（新疆大学数学与系统科学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
In this paper we study fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. To prepare the ground, we define various types of subhypergraphs, as well as various types of walks in a hypergraph. We then prove a number of new results involving cut edges, cut vertices, and blocks. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.
For any finite abelian group $G$ and any subset $S\seq G$, we determine the connectivity of the addition Cayley graph induced by $S$ on $G$. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form.
For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a monochromatic edge-cut. For a connected graph $G$, the monochromatic disconnection number (or $MD$-number for short) of $G$, denoted by $md(G)$, is the maximum number of colors that are allowed in order to make $G$ monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their $MD$-numbers are $1$. For graphs with diameter at leas…
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bo…
\noindent The b-chromatic number of a graph $G$, denoted by $ϕ(G)$, is the largest integer $k$ that $G$ admits a proper coloring by $k$ colors, such that each color class has a vertex that is adjacent to at least one vertex in each of the other color classes. El Sahili and Kouider [About b-colorings of regular graphs, Res. Rep. 1432, LRI, Univ. Orsay, France, 2006] asked whether it is true that every $d$-regular graph $G$ of girth at least 5 satisfies $ϕ(G)=d+1$. Blidia, Maffray, and Zemir [On b-colorings in regular graphs, Discrete Appl. Math. 157 (2009), 1787-1793] showed that the Petersen g…
In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness [A Menger-like property of tree-width: The finite case. Journal of Combinatorial Theory, Series B, 48(1):67-76, 1990]. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan as a possible edge-version of tree decompositions [The structure of graphs not admitting a fixed immersion. Journal of Combinatorial Theory, Se…
For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $κ^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $κ^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$…
The dual of a polyhedron is a polyhedron -- or in graph theoretical terms: the dual of a 3-connected plane graph is a 3-connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity of the dual on higher surfaces. Are the duals of 3-connected embedded graphs of higher genus 3-connected, too? If not: which connectivity guarantees 3-connectedness of the dual? In this article, we give answers to this and related questions. Among other things, we prove that there is no connectivity that for every genus guarantees the 3-connectedness or 2-co…
In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple $O(κn)$-space data structure that can quickly answer pairwise $(κ+1)$-connectivity queries in a $κ$-connected graph. We also show how to compute the "closest" $κ$-cut to every vertex in near linear $\tilde{O}(m+poly(κ)n)$ time.
The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesník (1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesník by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(…
Testing a graph on 2-vertex- and 2-edge-connectivity are two fundamental algorithmic graph problems. For both problems, different linear-time algorithms with simple implementations are known. Here, an even simpler linear-time algorithm is presented that computes a structure from which both the 2-vertex- and 2-edge-connectivity of a graph can be easily "read off". The algorithm computes all bridges and cut vertices of the input graph in the same time.
The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $κ_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum …
In this article, we give conditions on a graph under which the Lovász' original bound of the graph can be improved by increasing the topological connectivity of its neighbourhood complex. We also work out conditions under which computing the topological connectivity of hom complex of a pair of graphs can be simplified. In particular, hom complex as a covariant functor acting on a double mapping cylinder of graphs is a homotopy pushout of hom complex functor applied to its subgraphs. We give applications of this result where the computation of hom complexes is simplified. Finally, we explain wh…
The $\ell$-component connectivity (or $\ell$-connectivity for short) of a graph $G$, denoted by $κ_\ell(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This generalization is a natural extension of the classical connectivity defined in term of minimum vertex-cut. As an application, the $\ell$-connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability a…
Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will…
We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discrimin…
Reliability evaluation and fault tolerance of an interconnection network of some parallel and distributed systems are discussed separately under various link-faulty hypotheses in terms of different $\mathcal{P}$-conditional edge-connectivity. With the help of edge isoperimetric problem's method in combinatorics, this paper mainly offers a novel and unified view to investigate the $\mathcal{P}$-conditional edge-connectivities of hamming graph $K_{L}^{n}$ with satisfying the property that each minimum $\mathcal{P}$-conditional edge-cut separates the $K_{L}^{n}$ just into two components, such as …
Given a graph $G$, an arithmetical structure on $G$ is a pair of positive integer vectors $({\bf d},{\bf r})$ such that $\mathrm{gcd}({\bf r}_v\, | \,v\in V(G))=1$ and \[ (\mathrm{diag}({\bf d})-A){\bf r}=0, \] where $A$ is the adjacency matrix of $G$. We describe the arithmetical structures on graph $G$ with a cut vertex $v$ in terms of the arithmetical structures on their blocks. More precisely, if $G_1,\ldots,G_s$ are the induced subgraphs of $G$ obtained from each of the connected components of $G-v$ by adding the vertex $v$ and their incident edges, then the arithmetical structures on $G$…
For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $κ_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed…
