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Showing papers on "Path graph published in 2007"


Journal ArticleDOI
TL;DR: Here, thanks to a “principle of deferred decisions,” the percolation dynamics is described by a surprisingly simple Markov chain, which is replaced by a deterministic dynamical system, and its integrals are used to show—via exponential supermartingales—that thePercolation process undergoes relatively small fluctuations around the deterministic trajectory.
Abstract: The k-parameter bootstrap percolation on a graph is a model of an interacting particle system, which can also be viewed as a variant of a cellular automaton growth process with threshold k ≥ 2. At the start, each of the graph vertices is active with probability p and inactive with probability 1 − p, independently of other vertices. Presence of active vertices triggers a bootstrap percolation process controlled by a recursive rule: an active vertex remains active forever, and a currently inactive vertex becomes active when at least k of its neighbors are active. The basic problem is to identify, for a given graph, p− and p+ such that for p p+ resp.) the probability that all vertices are eventually active is very close to 0 (1 resp.). The bootstrap percolation process is a deterministic process on the space of subsets of the vertex set, which is easy to describe but hard to analyze rigorously in general. We study the percolation on the random d-regular graph, d ≥ 3, via analysis of the process on the multigraph counterpart of the graph. Here, thanks to a “principle of deferred decisions,” the percolation dynamics is described by a surprisingly simple Markov chain. Its generic state is formed by the counts of currently active and nonactive vertices having various degrees of activation capabilities. We replace the chain by a deterministic dynamical system, and use its integrals to show—via exponential supermartingales—that the percolation process undergoes relatively small fluctuations around the deterministic trajectory. This allows us to show existence of the phase transition within an interval [p−(n),p+(n)], such that (1) p±(n) p* = 1 − miny∈(0,1)y/ℙ(Bin(d − 1,1 − y) < k); (2) p+(n) − p−(n) is of order n−1/2 for k < d − 1, and n, (en 0,en log n ∞), for k = d − 1. Note that p* is the same as the critical probability of the process on the corresponding infinite regular tree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 30, 257–286, 2007

204 citations


Journal ArticleDOI
TL;DR: In this paper, lower and upper bounds for the Estrada index of G are established in terms of the number of vertices and number of edges of the graph and some inequalities between EE and the energy of G were obtained.

190 citations


Journal IssueDOI
TL;DR: The size of the t-spanner computed essentially matches the worst case lower bound implied by a 43-year old girth lower bound conjecture made independently by Erdos, Bollobas, and Bondy & Simonovits.
Abstract: Let G = (V,E) be an undirected weighted graph on |V | = n vertices and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V,ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well-motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43-year old girth lower bound conjecture made independently by Erdos, Bollobas, and Bondy & Simonovits. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information, which involves growing either breadth first search trees up to t(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 Preliminary version of this work appeared in the 30th International Colloquium on Automata, Languages and Programming, pages 384–396, 2003.

159 citations


Journal Article
TL;DR: In this paper, the authors considered the on-line shortest path problem under various models of partial monitoring and gave an algorithm with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant).
Abstract: The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/√n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant) and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m ≪ n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

136 citations


Journal ArticleDOI
TL;DR: It is shown that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices, and every graph with minimal degree at least 4 has total domination number at most 3n/7.
Abstract: The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvatal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

115 citations


Journal ArticleDOI
TL;DR: The maximum number of cliques in a graph for the following graph classes is determined: graphs with n vertices and m edges, d-degenerate graphs, and planar graphs.
Abstract: A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K5-minor or no K3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2).

81 citations


Journal ArticleDOI
TL;DR: If m>=2 then whp the cover time of a simple random walk on G"m(n) is asymptotic to 2mm-1nlogn, which has been proposed as a simple model of the world wide web.

78 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for fixed d ≥ 2 and 0 < e < 1, there exists a constant c = c(d, e) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − e)n vertices with maximum degree at most d.
Abstract: We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − e)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < e < 1, there exists a constant c = c(d, e) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − e)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and e, then G has a copy of every tree T as above.

76 citations


Journal ArticleDOI
TL;DR: It is established in particular that a minimum r-identifying code contains at least @?log"2(n+1)@?

74 citations


Journal Article
TL;DR: In this article, the first and second Zagreb indices are defined as [GRAPHICS], and it is shown that for all chemical graphs M-1/n <= M-2/m, G is a chemical graph.
Abstract: Let G = (V, E) be a simple graph with n = vertical bar V vertical bar vertices and m = vertical bar E vertical bar edges ; let d(1), d(2), ..., d(n) denote the degrees of the vertices of G. If Delta = max d(i) <= 4, G is a chemical graph. The first and second Zagreb indices are defined as [GRAPHICS] We show that for all chemical graphs M-1/n <= M-2/m. This does not hold for all general graphs, connected or not.

64 citations


Journal ArticleDOI
TL;DR: It is shown that the longest path problem can be solved efficiently for some tree-like graph classes by this approach, and two new graph classes that have natural interval representations are proposed.
Abstract: The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

Journal ArticleDOI
01 Sep 2007
TL;DR: It is shown that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique, and an algorithm is proposed that generates all non-induced bicliques of a graphs.
Abstract: An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B=X∪Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X,Y≠O, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. When the requirement that X and Y are independent sets of G is dropped, we have a non-induced biclique. We show that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique. We propose an algorithm that generates all non-induced bicliques of a graph. In addition, we propose specialized efficient algorithms for generating the bicliques of special classes of graphs.

Journal ArticleDOI
TL;DR: An algorithm is derived which decides in time O(m + n + 5.88k) whether a given graph with m edges and n vertices admits an LA of cost at most m + k (the algorithm computes such an LA if it exists), based on a procedure which generates a problem kernel of linear size in linear time for a connected graph G.
Abstract: A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph G admits an LA of cost bounded by a given integer. Since every edge of G contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT "parameterized above guaranteed value." We answer this question positively by deriving an algorithm which decides in time O(m + n + 5.88k) whether a given graph with m edges and n vertices admits an LA of cost at most m + k (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph G. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P = NP.

Proceedings ArticleDOI
11 Jun 2007
TL;DR: This work presents the first truly sub-cubic algorithm for APBP in general dense graphs and gives a procedure for computing the (max, min)-product of two arbitrary matrices over R ∪ (∞,-∞) in O(n2+Ω/3) ≤ O( n2.792) time.
Abstract: In the all-pairs bottleneck paths (APBP) problem (a.k.a. all-pairs maximum capacity paths), one is given a directed graph with real non-negative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all-pairs shortest paths.We present the first truly sub-cubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max, min)-product of two arbitrary matrices over R ∪ (∞,-∞) in O(n2+Ω/3) ≤ O(n2.792) time, where n is the number of vertices and Ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.

Proceedings ArticleDOI
Liam Roditty1
07 Jan 2007
TL;DR: The first approximation algorithm for finding the k-simple shortest paths connecting a pair of vertices in a weighted directed graph is obtained.
Abstract: We obtain the first approximation algorithm for finding the k-simple shortest paths connecting a pair of vertices in a weighted directed graph. Our algorithm is deterministic and has a running time of O(k(m√n + n3/2 log n)) where m is the number of edges in the graph and n is the number of vertices. Let s, t e V; the length of the i-th simple path from s to t computed by our algorithm is at most 3/2 times the length of the i-th shortest simple path from s to t. The best algorithms for computing the exact k-simple shortest paths connecting a pair of vertices in a weighted directed graph are due to Yen [19] and Lawler [13]. The running time of their algorithms, using modern data structures, is O(k(mn + n2 log n)). Both algorithms are from the early 70's. Although this problem and other variants of the k-shortest path problem drew a lot of attention during the last three and a half decades, the O(k(mn + n2 log n)) bound is still unbeaten.

Journal ArticleDOI
TL;DR: It is shown that to each graceful labelling of a path on 2s+1 vertices, s>=2, there corresponds a current assignment on a 3-valent graph which generates at least 2^2^s cyclic oriented triangular embeddings of a complete graph on 12s+7 vertices.

01 Jan 2007
TL;DR: The on-line shortest path problem is considered under various models of partial monitoring, and a version of the multi-armed bandit setting for shortest path is discussed, where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path.
Abstract: The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1= p n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant) and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of disconnecting a graph by removing as few vertices as possible, such that no component of the disconnected graph has more than a given number of vertices.
Abstract: In this paper, we consider the problem of disconnecting a graph by removing as few vertices as possible, such that no component of the disconnected graph has more than a given number of vertices. We give applications of this problem, present a formulation for it, and describe some polyhedral results.Furthermore, we establish ties with other polytopes and show how these relations can be used to obtain facets of our polytope. Finally, we give some computational results.

Journal ArticleDOI
TL;DR: It is proved that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f( k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite.
Abstract: We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Bohme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.

Journal ArticleDOI
TL;DR: A phase transition for the diameter is established when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3) and it is shown that for τ > 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is bounded from below by a constant times the logarithm of the size of the graph.
Abstract: In this paper, we study the configuration model (CM) with independent and identically-distributed (i.i.d.) degrees. We establish a phase transition for the diameter when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ > 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph.

Journal ArticleDOI
TL;DR: This paper proposes the first polynomial-time algorithm, which runs in O(|V|^9) time, to solve the path cover problem on distance-hereditary graphs.

Journal ArticleDOI
TL;DR: A polynomial algorithm in O(nm) time to find a long path in any graph with n vertices and m edges is obtained, bounded by a parameter defined on neighborhood condition of any three independent vertices of the path.

30 Jan 2007
TL;DR: New algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G are presented and a new method for measuring acyClicity based on modifications to two existing methods is presented.
Abstract: This paper presents new algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G. The first part introduces higher-order decomposition. This decomposition is an extension of the technique of strongly connected component (sc-component) decomposition. The second part presents a new method for measuring acyclicity based on modifications to two existing methods. In the new method, we decompose the graph into a 1-dominator set, which is a set of acyclic subgraphs where each subgraph is dominated by one trigger vertex. Meanwhile we compute sc-components of a degenerated graph derived from triggers. Using this preprocessing, a new SSSP algorithm has O(m + rlogl) time complexity, where r is the size of the 1-dominator set, and l is the size of the largest sc-component. In the third part, we modify the concept of a 1-dominator set to that of a 1-2-dominator set. Each of acyclic subgraphs obtained by the 1-2-dominator decomposition are dominated by one or two trigger vertices cooperatively. Such subgraphs are potentially larger than those decomposed by the 1-dominator set. Thus fewer trigger vertices are needed to cover the graph.

Book ChapterDOI
15 Aug 2007
TL;DR: Two lineartime algorithms for the weighted p-center problem for points on the real line are proposed, thereby partially resolving a long-standing open problem.
Abstract: An optimal linear time algorithm for the unweighted p-center problems in trees has been known since 1991 [4]. No such worst-case linear time result is known for the weighted version of the p-center problems, even for a path graph. In this paper, for fixed p, we propose two lineartime algorithms for the weighted p-center problem for points on the real line, thereby partially resolving a long-standing open problem. One of our approaches generalizes the trimming technique of Megiddo [10], and the other one is based on the parametric pruning technique, introduced here. The proposed solutions make use of the solutions of another variant of the center problem called the conditional center location problem [13].

Proceedings ArticleDOI
07 Jan 2007
TL;DR: This work shows that for any ε > 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(ω + 2+ε), where ω is the exponent of fastest matrix multiplication algorithm.
Abstract: We show that for any e > 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(nω + n2+e), where ω is the exponent of fastest matrix multiplication algorithm. By the currently best bound on ω, the running time of our algorithm is O(n2.376). Our algorithm substantially improves the previous time-bounds for this problem recently established by Vassilevska et al. (STOC 2006, O(n2.688)) and (ICALP 2006, O(n2.575)). Its asymptotic time complexity matches that of the fastest known algorithm for finding a triangle (not necessarily a maximum-weight one) in a graph.By applying or extending our algorithm, we can also improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph established in the papers by Vassilevska et al. For example, we can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time required by the fastest algorithm for finding any triangle in a graph with m edges, i.e., in time O(m1.41).

Journal ArticleDOI
TL;DR: Sharp upper bounds on the maximum number of edges in a bar $k$-visibility graph on $n$ vertices and the largest order of a complete bar $ k$- Visibility graph are presented.
Abstract: A bar visibility representation of a graph $G$ is a collection of horizontal bars in the plane corresponding to the vertices of $G$ such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar $k$-visibility graph, two vertices are adjacent if and only if the corresponding bars can be joined by a vertical line segment that intersects at most $k$ other bars. Bar $k$-visibility graphs were introduced by Dean et al. [J. Graph Algorithms Appl., 11 (2007), pp. 45-59]. In this paper, we present sharp upper bounds on the maximum number of edges in a bar $k$-visibility graph on $n$ vertices and the largest order of a complete bar $k$-visibility graph. We also discuss regular bar $k$-visibility graphs and forbidden induced subgraphs of bar $k$-visibility graphs.

12 Jan 2007
TL;DR: The commuting graph of a group G is a simple graph whose vertices are all non-central elements of G and two dis-tinct vertices x,y are adjacent if xy = yx.
Abstract: The commuting graph of a group G, denoted by i(G), is a simple graph whose vertices are all non-central elements of G and two dis- tinct vertices x,y are adjacent if xy = yx. In (1) it is conjectured that if M is a simple group and G is a group satisfying i(G) » i(M), then G » M. In this paper we prove this conjecture for many simple groups.

Journal ArticleDOI
TL;DR: It is proved that if a graph on n vertices has more than (k-1)n2 edges, then it contains every k-edge spider of 3 legs, and also, every k -edge spider with no leg of length more than 4, which strengthens a result of Wozniak on spiders of diameter at most 4.

Journal IssueDOI
TL;DR: In this paper, a lower bound on the degree of vertices and the vertex-degree of the ends which is quadratic in k, the connectedness of the desired subgraph is given.
Abstract: A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex-degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2k at the vertices and a vertex-degree of order k log k at the ends which have no k-connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge-degree for the ends (which is defined as the maximum number of edge-disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge-degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)-edge-connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007

Journal ArticleDOI
17 Dec 2007
TL;DR: It is proved that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree.
Abstract: It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.