Edge reductions in cyclically k-connected cubic graphs

AbstractThis paper examines edge reductions in cyclically k-connected cubic graphs, focusing on when they preserve the cyclic k-connectedness. For a cyclically k-connected cubic graph G, we denote by Nk(G) the set of edges whose reduction gives a cubic graph which is not cyclically k-connected. With the exception of three graphs, Nk(G) consists of the edges in independent k-edge cuts. For this reason we examine the properties and interactions between independent k-edge cuts in cyclically k-connected cubic graphs. These results lead to an understanding of the structure of G[Nk]. For every k, we prove that G[Nk] is a forest with at least k trees if G is a cyclically k-connected cubic graph with girth at least k + 1 and Nk ≠ ⊘. Let fk(ν) be the smallest integer such that |Nk(G)| ≤ fk(ν) for all cyclically k-connected cubic graphs G on ν vertices. For all cyclically 3-connected cubic graphs G such that 6 ≤ ν(G) and N3 ≠ ⊘, we prove that G[N3] is a forest with at least three trees. We determine f3 and state a characterization of the extremal graphs. We define a very restricted subset N4b of N4 and prove that if N4g = N4 − N4b ≠ ⊘, then G[N4g] is a forest with at least four trees. We determine f4 and state a characterization of the extremal graphs. There exist cyclically 5-connected cubic graphs such that E(G) = N5(G), for every ν such that 10 ≤ ν and 16 ≠ ν. We characterize these graphs. Let gk(ν) be the smallest integer such that |Nk(G)| ≤ gk(ν) for all cyclically k-connected cubic graphs G with ν vertices and girth at least k + 1. For k ∈ {3, 4, 5}, we determine gk and state a characterization of the extremal graphs

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure

Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time \poly(n) 2^{O(k)} due to Chlamt\'a\v{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a $\rho$-approximation for series-parallel graphs (where $\rho \geq 1$), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to $1/\rho$. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than $17/16 - \epsilon$ for $\epsilon > 0$; assuming the Unique Games Conjecture the hardness becomes $1/\alpha_{GW} - \epsilon$. For graphs with large (but constant) treewidth, we show a hardness result of $2 - \epsilon$ assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation

On the complexity of computing the kk-restricted edge-connectivity of a graph

The \emph{$k$-restricted edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least $k$ vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing $\lambda_k(G)$. Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the $k$-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.Comment: 16 pages, 4 figure

3-Factor-criticality of vertex-transitive graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if and only if it is not a cycle.Comment: 15 pages, 3 figure