Edge Deletion Research Articles

Inertia sets for graphs on six or fewer vertices

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G), a question which was previously answered when G is a tree. In this paper, a number of new techniques are developed in order to be able to determine possible inertias of general graphs: covers with cliques, covers with cliques and clique-stars, and the graph operations of edge subdivision, edge deletion, joins, and unions. Because most of the associated theorems require additional hypotheses, definitive criteria that apply to all graphs cannot be provided. Nevertheless, these results are strong enough to be able to determine the inertia set of each graph on 6 or fewer vertices and can be applied to many graphs with larger order as well. One consequence of the 1–6 vertex results is the fact that all of these graphs have balanced inertia. It is also mentioned which of these results guarantee or preserve balanced inertia, and explain how to modify them to include Hermitian matrices.

On the Degree Sequence of an Evolving Random Graph Process and Its Critical Phenomenon

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

On the Degree Sequence of an Evolving Random Graph Process and Its Critical Phenomenon

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

Graph transformations preserving the stability number

We analyse the relations between several graph transformations that were introduced to be used in procedures determining the stability number of a graph. We show that all these transformations can be decomposed into a sequence of edge deletions and twin deletions. We also show how some of these transformations are related to the notion of even pair introduced to color some classes of perfect graphs. Then, some properties of edge deletion and twin deletion are given and a conjecture is formulated about the class of graphs for which these transformations can be used to determine the stability number.

How divergence mechanisms influence disassortative mixing property in biology

The duplication–divergence mechanism of network growth has been widely investigated, especially in gene and protein networks. Both the duplication and divergence have a key role in biological network evolution. However, the relative roles of these mechanisms in the influence of disassortative property in protein interaction networks remain to be clarified. It has been shown that duplication can indeed make protein networks evolve towards disassortative networks. To make the relationship between the disassortative property and the duplication–divergence mechanism more clear, we further discuss how the divergence mechanism influences the disassortative property. We tested four different divergence mechanisms, i.e., node deletion, edge deletion, edge addition, and edge rewiring to study their effects on disassortative property. Our study highlights the crucial roles of different divergence evolution mechanisms.

Additive approximation for edge-deletion problems

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E' P (G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. • For any fixed e > 0 and any monotone property P, there is a deterministic algorithm which, given a graph G = (V, E) of size n, approximates E' P (G) in linear time O(|V| + |E|) to within an additive error of en 2 . Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E' P . Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. (1) If there is a bipartite graph that does not satisfy P, then there is a δ > 0 for which it is possible to approximate E' P to within an additive error of n 2 -δ in polynomial time. (2) On the other hand, if all bipartite graphs satisfy P, then for any δ > 0 it is NP-hard to approximate E' P to within an additive error of n 2-δ . While the proof of (1) is relatively simple, the proof of (2) requires several new ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing E' P precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis, who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E' P is NP-hard.

An extension of the bivariate chromatic polynomial

K. Dohmen, A. Pönitz and P. Tittmann [K. Dohmen, A. Pönitz, P. Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6 (2003), 69–90], introduced a bivariate generalization of the chromatic polynomial P ( G , x , y ) which subsumes also the independent set polynomial of I. Gutman and F. Harary [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematicae 24 (1983), 97–106] and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little [F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little, The vertex-cover polynomial of a graph, Discrete Mathematics 250 (2002), 71–78]. We first show that P ( G , x , y ) has a recursive definition with respect to three kinds of edge eliminations: edge deletion, edge contraction, and edge extraction, i.e. deletion of an edge together with its endpoints. Like in the case of deletion and contraction only [J.G. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: J.A. Bundy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 329–339] it turns out that there is a most general, or as they call it, a universal polynomial satisfying such recurrence relations with respect to the three kinds of edge eliminations, which we call ξ ( G , x , y , z ) . We show that the new polynomial simultaneously generalizes, P ( G , x , y ) , as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of ξ ( G , x , y , z ) using a subset expansion formula. We also show that ξ ( G , x , y , z ) can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky [T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992), 317–347] and by B. Bollobás and O. Riordan [B. Bollobás, O. Riordan, A Tutte polynomial for coloured graphs, Combinatorics, Probability and Computing 8 (1999), 45–94]. The edge-labeled polynomial ξ lab ( G , x , y , z , t ̄ ) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. [R.C. Read, E.G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Mathematics 204 (1999), 337–356]. Finally, we discuss the complexity of computing ξ ( G , x , y , z ) .

Growing networks with preferential deletion and addition of edges

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t = 1 , 2 , … , with probability π 1 > 0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π 2 a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π 3 = 1 − π 1 − π 2 < 1 / 2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p k of vertices with degree k , and solving this recursion reveals that, for π 3 < 1 / 3 , we have p k ∼ k − ( 3 − 7 π 3 ) / ( 1 − 3 π 3 ) , while, for π 3 > 1 / 3 , the fraction p k decays exponentially at rate ( π 1 + π 2 ) / 2 π 3 . There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.

Balanced cut approximation in random geometric graphs

A random geometric graph G ( n , r ) is obtained by spreading n points uniformly at random in a unit square, and by associating a vertex to each point and an edge to each pair of points at Euclidian distance at most r . Such graphs are extensively used to model wireless ad-hoc networks, and in particular sensor networks. It is well known that, over a critical value of r , the graph is connected with high probability. In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges. In particular, we show that, for a sufficiently large r , any cut which separates two components of Θ ( n ) vertices each contains Ω ( n 2 r 3 ) edges with high probability. We also present a simple algorithm that, again with high probability, computes one such cut of size O ( n 2 r 3 ) . From these two results we derive a constant expected approximation algorithm for the β -balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least β n vertices each.

Applying path‐counting methods for measuring the robustness of the network‐structured system

AbstractWe consider the path‐counting problem that asks how many paths exist between any two different nodes in a network after deleting an arbitrary number of edges or nodes from the original network. Using path‐counting methods, we propose a quantitative method for measuring the robustness of the network‐structured system. First we define the connectivity function and attempt to obtain the expected connectivity function for the network. Applying the Monte Carlo method, we estimate expected edge deletion and node deletion connectivity functions when an arbitrary number of edges or nodes are deleted from the original network. We attempt to approximate the expected edge deletion connectivity function using an appropriate nonlinear function with two parameters. We also show the numerical results of applying the path‐counting method with their analysis in order to quantitatively evaluate the connectivity for some special types of networks.

A Fully Dynamic Graph Algorithm for Recognizing Interval Graphs

We present the first dynamic graph algorithm for recognizing interval graphs. The algorithm runs in O(nlog n) worst-case time per edge deletion or edge insertion, where n is the number of vertices in the graph. The algorithm uses a new representation of interval graphs called the train tree, which is based on the clique-separator graph representation of chordal graphs. The train tree has a number of useful properties and it can be constructed from the clique-separator graph in O(n) time.

On Generating Triangle-Free Graphs

We show that the problem to decide whether a graph can be made triangle-free with at most k edge deletions remains NP-complete even when restricted to planar graphs of maximum degree seven. In addition, we provide polynomial-time data reduction rules for this problem and obtain problem kernels consisting of 6k vertices for general graphs and 11k/3 vertices for planar graphs.

On percolation in random graphs with given vertex degrees

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.

Unavoidable parallel minors of 4‐connected graphs

AbstractA parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4‐connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k‐partition triple fan with a complete graph on the vertices of degree k, the k‐spoke double wheel, the k‐spoke double wheel with axle, the (2k+1)‐rung Möbius zigzag ladder, the (2k)‐rung zigzag ladder, or Kk. We also find the unavoidable parallel minors of 1‐, 2‐, and 3‐connected graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 313‐326, 2009

Vertex Separators of Graphs

Let G = ( V , E ) be a connected simple graph. A non-empty subset C of V is a separator of G if there exist two non-empty subsets A and B of V such that { A , B , C } is a partition of V , max{| A |, | B |} ≤ ½ | V |, and no edge of G joins a vertex in A and a vertex in B . If G has a separator, its separability sep( G ) is the cardinality of a minimum separator of G . This paper characterizes singleton and doubleton separators of graphs. It determines the separabilities of some special graphs, sum of graphs, and graphs resulting from deletion of edges.

A randomized algorithm for the joining protocol in dynamic distributed networks

We describe a randomized algorithm for assigning neighbours to vertices joining a dynamic distributed network. The aim of the algorithm is to maintain connectivity, low diameter and constant vertex degree. On joining each vertex donates a constant number of tokens to the network. These tokens contain the address of the donor vertex. The tokens make independent random walks in the network. A token can be used by any vertex it is visiting to establish a connection to the donor vertex. This allows joining vertices to be allocated a random set of neighbours although the overall vertex membership of the network is unknown. The network we obtain in this way is robust under adversarial deletion of vertices and edges and actively reconnects itself. One model we consider is a network constructed in this fashion, in which vertices join but never leave. If t is the size of the network, then the diameter of the network is O ( log t ) for all t , with high probability. As an example of the robustness of this model, suppose an adversary deletes edges from the network leaving components of size at least t 1 / 2 + δ . With high probability the network reconnects itself by replacing lost edges using tokens from the token pool.

On enumeration of conjugacy classes of Coxeter elements

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given.

A Multivariate Interlace Polynomial and its Computation for Graphs of Bounded Clique-Width

We define a multivariate polynomial that generalizes in a unified way the two-variable interlace polynomial defined by Arratia, Bollobás and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other. We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte's polynomial and of its multivariate generalizations are based on edge deletions and contractions. We also show that bounded portions of our polynomial can be evaluated in polynomial time for graphs of bounded clique-width. Our proof uses an expression of the interlace polynomial in monadic second-order logic, and works actually for every polynomial expressed in monadic second-order logic in a similar way.

Diameter vulnerability of graphs by edge deletion

Let f ( t , k ) be the maximum diameter of graphs obtained by deleting t edges from a ( t + 1 ) -edge-connected graph with diameter k . This paper shows 4 2 t − 6 < f ( t , 3 ) ≤ max { 59 , 5 2 t + 7 } for t ≥ 4 , which corrects an improper result in [C. Peyrat, Diameter vulnerability of graphs, Discrete Appl. Math. 9 (3) (1984) 245–250] and also determines f ( 2 , k ) = 3 k − 1 and f ( 3 , k ) = 4 k − 2 for k ≥ 3 .

On Identifying Codes in the King Grid that are Robust Against Edge Deletions

Assume that $G = (V, E)$ is an undirected graph, and $C \subseteq V$. For every $v \in V$, we denote $I_r(G;v) = \{ u \in C: d(u,v) \leq r\}$, where $d(u,v)$ denotes the number of edges on any shortest path from $u$ to $v$. If all the sets $I_r(G;v)$ for $v \in V$ are pairwise different, and none of them is the empty set, the code $C$ is called $r$-identifying. If $C$ is $r$-identifying in all graphs $G'$ that can be obtained from $G$ by deleting at most $t$ edges, we say that $C$ is robust against $t$ known edge deletions. Codes that are robust against $t$ unknown edge deletions form a related class. We study these two classes of codes in the king grid with the vertex set ${\Bbb Z}^2$ where two different vertices are adjacent if their Euclidean distance is at most $\sqrt{2}$.