42,696 research outputs found
Computing the Largest Bond of a Graph
A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S,VS] of G such that G[S] and G[VS] are both connected. Given s,t in V(G), an st-bond of G is a bond whose removal disconnects s and t. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest st-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that Largest Bond remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless P = NP. We also show that Largest Bond and Largest st-Bond on graphs of clique-width w cannot be solved in time f(w) x n^{o(w)} unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) x n^{O(w)}. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP subseteq coNP/poly
Intersection of Longest Cycle and Largest Bond in 3-Connected Graphs
A bond in a graph is a minimal nonempty edge-cut. A connected graph is
dual Hamiltonian if the vertex set can be partitioned into two subsets and
such that the subgraphs induced by and are both trees. There is
much interest in studying the longest cycles and largest bonds in graphs. H. Wu
conjectured that any longest cycle must meet any largest bond in a simple
3-connected graph. In this paper, the author proves that the above conjecture
is true for certain classes of 3-connected graphs: Let be a simple
3-connected graph with vertices and edges. Suppose is the size
of a longest cycle, and is the size of a largest bond. Then each
longest cycle meets each largest bond if either or . Sanford determined in her Ph.D. thesis the cycle spectrum of
the well-known generalized Petersen graph ( is odd) and
( is even). Flynn proved in her honors thesis that any generalized Petersen
graph is dual Hamiltonian. The author studies the bond spectrum
(called the co-spectrum) of the generalized Petersen graphs and extends Flynn's
result by proving that in any generalized Petersen graph , , the co-spectrum of is .Comment: 16 pages, 19 figures. Paper presented at the 54th Southeastern
International Conference on Combinatorics, Graph Theory and Computing (March
6-10, 2023); submitted on May 9, 2023 to the conference proceedings book
series publication titled "Springer Proceedings in Mathematics and
Statistics" (PROMS). Paper abstract also on
https://www.math.fau.edu/combinatorics/abstracts/ren54.pd
Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices
Percolation threshold of a network is the critical value such that when nodes
or edges are randomly selected with probability below the value, the network is
fragmented but when the probability is above the value, a giant component
connecting large portion of the network would emerge. Assessing the percolation
threshold of networks has wide applications in network reliability, information
spread, epidemic control, etc. The theoretical approach so far to assess the
percolation threshold is mainly based on spectral radius of adjacency matrix or
non-backtracking matrix, which is limited to dense graphs or locally treelike
graphs, and is less effective for sparse networks with non-negligible amount of
triangles and loops. In this paper, we study high-order non-backtracking
matrices and their application to assessing percolation threshold. We first
define high-order non-backtracking matrices and study the properties of their
spectral radii. Then we focus on 2nd-order non-backtracking matrix and
demonstrate analytically that the reciprocal of its spectral radius gives a
tighter lower bound than those of adjacency and standard non-backtracking
matrices. We further build a smaller size matrix with the same largest
eigenvalue as the 2nd-order non-backtracking matrix to improve computation
efficiency. Finally, we use both synthetic networks and 42 real networks to
illustrate that the use of 2nd-order non-backtracking matrix does give better
lower bound for assessing percolation threshold than adjacency and standard
non-backtracking matrices.Comment: to appear in proceedings of the 26th International World Wide Web
Conference(WWW2017
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