Oriented ribbon graphs (dessins d’enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L , there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring L . This corresp...

We provide the rst nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NP-Hard problem of nding a heaviest planar subgraph in an edge-weighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that pro...

We present a general framework for obtaining efficient algorithms for computing minimum spanning trees. We use this framework to derive the classical algorithms of Prim, Kruskal and Bor̊uvka. We then describe the randomized linear-time algorithm of Karger, Klein and Tarjan. The algorithm of Karger, Klein and Tarjan uses deterministic linear-time implementations of a verification algorithm of Kom...

We consider a variation of the chip-firing game in an induced subgraph S of a graph G. Starting from a given chip configuration, if a vertex v has at least as many chips as its degree, we can fire v by sending one chip along each edge from v to its neighbors. Chips are removed at the boundary δS. The game continues until no vertex can be fired. We will give an upper bound, in terms of Dirichlet...

Every chordal graph G admits a representation as the intersection graph of a family of subtrees of a tree. A classic way of finding such an intersection model is to look for a maximum spanning tree of the valuated clique graph of G. Similar techniques have been applied to find intersection models of chordal graph subclasses as interval graphs and path graphs. In this work, we extend those metho...

We prove that for fixed k, every k-uniform hypergraph on n vertices and of minimum codegree at least \(n/2+o(n)\) contains spanning tight k-tree bounded vertex degree as a subgraph. This generalises well-known result Komlos, Sarkozy Szemeredi graphs. Our is asymptotically sharp .

Online graph problems are considered in models where the irrevocability requirement is relaxed. We consider Late Accept model, a request can be accepted at later point, but any acceptance irrevocable. Similarly, we Reject an rejected, rejection irrevocable (this sometimes called preemption). Finally, Accept/Reject late accepts and rejects both allowed, reject four classical problems: For Maximu...

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider this problem in the setting of local algorithms: one wants to quickly determine whether a given edge e is in a specific spanning tree, without computing the whole spanning tree, but rather by inspecting the local neighborhood of e. The challenge is to maintain consistency. That is, to answer quer...