Abstract. In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [13] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O ( p g/n) and vertex bisectors of size O ( √ gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. 1. Introduction. Spectral

The parameter contraction degeneracy — the maximum minimum degree over all minors of a graph — is a treewidth lower bound and was first defined in [3]. In experiments it was shown that this lower bound improves upon other treewidth lower bounds [3]. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, blocks of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.

A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...

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