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An analysis of graph cut size for transductive learning
 In In proceedings of the 23rd International Conference on Machine Learning (ICML
, 2006
"... I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (r ..."
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I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (random) subset of the vertices are revealed to the learner.
Hyperbolicity and chordality of a graph
 2p + ⌊ε/2⌋, and d = 2p + ⌊ε/2⌋ + 1, which gives S1 = 2, S2 = 4p + 2 ⌊ε/2⌋, and so h(a, b, c, d) = ε − 2 − 2 ⌊ε/2⌋  ≤ 2. Let us now assume that S2 = max {S1, S2, S3}. Since S1 + S3 = 4p + ε, the
"... Let G be a connected graph with the usual shortestpath metric d. The graph G is δhyperbolic provided for any vertices x,y,u,v in it, the two larger of the three sums d(u,v) + d(x,y),d(u,x) + d(v,y) and d(u,y) + d(v,x) differ by at most 2δ. The graph G is kchordal provided it has no induced cycle ..."
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Let G be a connected graph with the usual shortestpath metric d. The graph G is δhyperbolic provided for any vertices x,y,u,v in it, the two larger of the three sums d(u,v) + d(x,y),d(u,x) + d(v,y) and d(u,y) + d(v,x) differ by at most 2δ. The graph G is kchordal provided it has no induced cycle of length greater than k. Brinkmann, Koolen and Moulton find that every 3chordal graph is 1hyperbolic and that graph is not 1hyperbolic if and only if it contains one of two special graphs 2 as an isometric subgraph. For every k ≥ 4, we show that a kchordal graph must be ⌊ k 2 ⌋ k−2
MATCHINGS, CONNECTIVITY, AND EIGENVALUES IN REGULAR GRAPHS
, 2011
"... We study extremal and structural problems in regular graphs involving various parameters. In Chapter 2, we obtain the best lower bound for the matching number over nvertex connected regular graphs in terms of edgeconnectedness and determine when the matching number is minimized. We also establish ..."
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We study extremal and structural problems in regular graphs involving various parameters. In Chapter 2, we obtain the best lower bound for the matching number over nvertex connected regular graphs in terms of edgeconnectedness and determine when the matching number is minimized. We also establish the best upper bound for the number of cutedges over nvertex connected odd regular graphs and determine when the number of cutedges is maximized. In addition, there is a relationship between the matching number and the total domination number in regular graphs. In Chapter 3, we explore the relationship between eigenvalue and matching number in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a ledgeconnected dregular graph, when l ≤ d − 2. We also study what is the weakest hypothesis on the second largest eigenvalue λ2 for a dregular graph G to guarantee that G is ledgeconnected. In Chapter 4, we study several extremal problems for regular graphs, including the Chinese postman problem, the path cover number, the average edgeconnectivity, and the number of perfect matchings. In Chapter 5, we study an rdynamic coloring problem and give the relationship between the rdynamic chromatic number and the chromatic number in regular graphs. We also study rdynichromatic number of the cartesian product of paths and cycles.
IMPROVED BOUNDS AND ALGORITHMS FOR GRAPH CUTS AND NETWORK RELIABILITY‡
"... Abstract. Karger (SIAM Journal on Computing, 1999) developed the first fullypolynomial approximation scheme to estimate the probability that a graph G becomes disconnected, given that its edges are removed independently with probability p. This algorithm runs in O(n5+o(1)−3) time to obtain an estim ..."
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Abstract. Karger (SIAM Journal on Computing, 1999) developed the first fullypolynomial approximation scheme to estimate the probability that a graph G becomes disconnected, given that its edges are removed independently with probability p. This algorithm runs in O(n5+o(1)−3) time to obtain an estimate within relative error . We improve this runtime in two key ways, one algorithmic and one graphtheoretic. From an algorithmic point of view, there is a certain key subproblem encountered by Karger, for which a generic estimation procedure is employed. We show that this subproblem has a special structure for which a much more efficient algorithm can be used. From a graphtheoretic point of view, we show better bounds on the number of edge cuts which are likely to fail. Karger’s analysis depends on bounds for various graph parameters; we show that these bounds cannot be simultaneously tight. We describe a new graph parameter, which simultaneously influences all the bounds used by Karger, and use it to obtain much tighter estimates of the behavior of the cuts of G. These techniques allow us to improve the runtime to O(n3+o(1)−2); our results also rigorously prove certain experimental observations of Karger & Tai (Proc. ACMSIAM Symposium on Discrete Algorithms, 1997). Our proofs, while rigorous, are motivated by certain nonrigorous differentialequation approximations which, however, track the worstcase trajectories of the relevant parameters. A key driver of Karger’s approach (and other cutrelated results) is his earlier bound on the number of small cuts: we also show how to improve this when the mincut size is “small ” and odd, augmenting, in part, a result of Bixby (Bulletin of the AMS, 1974). 1. Introduction. Let
Bipartite Powers of kchordal Graphs
, 2013
"... Let k be an integer and k ≥ 3. A graph G is kchordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G m is chordal then so is G m+2. B ..."
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Let k be an integer and k ≥ 3. A graph G is kchordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G m is chordal then so is G m+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchettype theorems for powers of HHDfree graphs. Discrete Mathematics, 177(13):916, 1997.] showed that if G m is kchordal, then so is G m+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The mth bipartite power G [m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u, v) where dG(u, v) is odd and less than or equal to m. Note that G [m] = G [m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is kchordal then so is G [m] , where k, m are positive integers with k ≥ 4.