Edge Separator Research Articles

Edge integrity of nearest neighbor graphs and separator theorems

We give upper bounds for the edge integrity and the vertex integrity of k-type nearest neighbor graphs in Rd. For n the order of a graph, examples are constructed to show that the bounds are best possible in the parameters n and k. The results follow from decompositions that are shown for certain graphs that possess small edge or vertex separators. A vertex separator theorem for nearest neighbor graphs is known, and an edge separator theorem is shown. We also apply the decompositions to obtain upper bounds for the edge integrity of planar graphs, graphs of fixed genus, and trees.

Partitioning a graph into small pieces with applications to path transversal

Given a graph $$G = (V, E)$$ and an integer $$k \in \mathbb {N}$$, we study $$k$$-Vertex Separator (resp. $$k$$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has less than k vertices. We also study $$k$$-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. Our main results are the following improved approximation algorithms.

Drawing Planar Graphs with Reduced Height

A polyline (resp., straight-line) drawing $\Gamma$ of a planar graph $G$ on a set $L_k$ of $k$ parallel lines is a planar drawing that maps each vertex of $G$ to a distinct point on $L_k$ and each edge of $G$ to a polygonal chain (resp., straight line segment) between its corresponding endpoints, where the bends lie on $L_k$. The height of $\Gamma$ is $k$, i.e., the number of lines used in the drawing. In this paper we establish new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every $n$-vertex planar graph with maximum degree $\Delta$, having an edge separator of size $\lambda$, admits a polyline drawing with height $4n/9+O(\lambda)$, where the previously best known bound was $2n/3$. Since $\lambda\in O(\sqrt{n\Delta})$, this implies the existence of a drawing of height at most $4n/9+o(n)$ for any planar triangulation with $\Delta \in o(n)$. For $n$-vertex planar $3$-trees, we compute straight-line drawings, with height $4n/9+O(1)$, which improves the previously best known upper bound of $n/2$. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the graph.

Continuous quadratic programming formulations of optimization problems on graphs

Four NP-hard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.

Optimality conditions for maximizing a function over a polyhedron

We present new first and second-order optimality conditions for maximizing a function over a polyhedron. These conditions are expressed in terms of the first and second-order directional derivatives along the edges of the polyhedron, and an edge description of the polyhedron. If the objective function is quadratic and edge-convex, and the constraint polyhedron includes a box constraint, then local optimality can be checked in polynomial time. The theory is applied to continuous formulations of the vertex and edge separator problems. It is seen that the necessary and sufficient optimality conditions for these problems are related to the existence of edges at specific locations in the graph.

Experimental study of whistle noise generated by insert edge and oil separator model of refrigeration cycle

The compressed air experiments are conducted to investigate the whistle noise radiated from the oil separator component of refrigerant cycle system. Two types of insert edge and a height-adjustable oil separator model are adopted. The acoustic characteristic of flow through plain top insert edge is more likely to be broadband. Flow through ramp top insert edge would induce whistle peaks at Mach number higher than 0.1197 and the oil separator model works like an acoustic amplifier. Moreover, the Strouhal number jumps are captured inside oil separator model and two mechanisms are presented to clarify the phenomena. First, the whistle noise at Modes A1, A3, B1 and B3 is regarded as coupling of shear layer instability with resonance acoustic modes of oil separator model. Second, the whistle noise at Modes A2 and B2 is regarded to be the feedback loop of flow–acoustic interaction in fluctuated shear layer and defined as jet–cavity interaction tones.

NP-completeness of the Planar Separator Problems

For a given graph G, the Separator Problem asks whether a vertex or edge set of small cardinality (or weight) exists whose removal partitions G into two disjoint graphs of approximately equal sizes. Called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set, both problems are NP-complete for general unweighted graphs [6]. Despite the significance of planar graphs, it has not been known whether the Planar Separator Problem, which considers a planar graph and a threshold as an input, is NP-complete or not. In this paper, we prove that the Vertex Separator Problem is in fact NP-complete when G is a vertex weighted planar graph. The Edge Separator Problem will be shown NP-complete when G is a vertex and edge weighted planar graph. In addition, we consider how to treat the constant � 2 R

Edge separators for graphs of bounded genus with applications

We prove that every n-vertex graph of genus g and maximal degree k has an edge separator of size O(√ gkn). The upper bound is best possible to within a constant factor. This extends the known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.

Edge Separators of Planar and Outerplanar Graphs With Applications

We consider the problem of finding a small set of edges (an edge separator) whose removal divides a given graph into roughly equal parts. We show that every planar graph with n vertices and a maximum degree k has an O([formula])-edge separator, which bound is asymptotically optimal. This improves the previously best known upper bound of O( k[formula]). We prove that the bisection width of the class of planar graphs and trees of n vertices and maximum degree k is O([formula] and O( k log n/log k), respectively, both optimal within a constant factor. All proofs are constructive and yield optimal O( n) algorithms. The separator theorems are applied to embed efficiently planar and outerplanar graphs into binary trees, hypercubes, butterfly graphs, CCC graphs, shuffle-exchange graphs, de Brujn graphs, and d-dimensional meshes. The embeddings are shown to be optimal or almost optimal with respect to the average dilation and the expansion of the embedding.

Partitioning Sparse Matrices with Eigenvectors of Graphs

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...