Edge separators for graphs of bounded genus with applications

$n$-vertex graph of positive genus $g$ and maximal degree $k$ has an $O(\sqrt{gkn})$ edge separator. This bound is best possible to within a constant factor. The separator can be found in $O(g+n)$ time provided that we start with an imbedding of the graph in its genus surface. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric problem, to efficient embeddings of graphs of genus $g$ into various classes of graphs including trees, meshes and hypercubes and to showing lower bounds on crossing numbers of $K_n,K_{m,n}$ and $Q_n$ drawn on surfaces of genus $g$

Edge separators for graphs excluding a minor

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sykora, and Vr\v{t}o (1993) for planar graphs, and of Sykora and Vr\v{t}o (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$

Edge separators for graphs of bounded genus with applications

SIGLEAvailable from TIB Hannover: RR1912(91-125) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Edge separators for graphs of bounded genus with applications

SIGLEAvailable from TIB Hannover: RR1912(91-125) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

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 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