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We show that, for every n and every surface $\Sigma$, there is a graph U embeddable on $\Sigma$ with at most cn^2 vertices that contains as minor every graph embeddable on $\Sigma$ with n vertices. The constant c depends polynomially on the Euler genus of $\Sigma$. This generalizes a well-known result for planar graphs due to Robertson, Seymour, and Thomas [Quickly Excluding a Planar Graph. J. Comb. Theory B, 1994] which states that the square grid on 4n^2 vertices contains as minor every planar graph with n vertices.

A graph is $O_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in $O_k$-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of $O_2$-free graphs without $K_{3,3}$-subgraph and whose treewidth is (at least) logarithmic. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $O_k$-free graphs, and that deciding the $O_k$-freeness of sparse graphs is polynomial time solvable., Comment: 29 pages, 6 figures. v4: updated intro

Given a property (graph class) $\Pi$, a graph $G$, and an integer $k$, the \emph{$\Pi$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $\Pi$ by adding at most $k$ edges to $G$. The $\Pi$-completion problem is known to be NP-hard for general graphs when $\Pi$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $\Pi$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class., Comment: 13 pages, 2 figures