If G is a geometric graph with n ≥ 5 vertices and for any set U with 5 vertices of G, the geometric subgraph of G, induced by U , has a plane spanning tree, then G has a plane spanning tree.

We present a 10 7 -approximation algorithm for the minimum two-vertex-connected spanning subgraph problem.

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgra...

Consider a set of n points in the plane, each one of which is colored either red, blue, or purple. A red-blue-purple spanning graph (RBP spanning graph) is a graph whose vertices are the points and whose edges connect the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. The minimum RBP spanning ...

Consider a set of n points in the plane, each one of which is colored either red, blue, or purple. A red-blue-purple spanning graph (RBP spanning graph) is a graph whose vertices are the points and whose edges connect the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. The minimum RBP spanning ...

A graph is chordal if every cycle of length greater than three contains an edge between non-adjacent vertices. Chordal graphs are of interest both theoretically, since they admit polynomial time solutions to a range of NP-hard graph problems, and practically, since they arise in many applications including sparse linear algebra, computer vision, and computational biology. A maximal chordal subg...

Proof. This involves proving P ⇔ Q where P is the statement T is a spanning tree of G and Q is the statement T has n − 1 edges. In both statements we are allowed to add on the fact that T is a spanning connected subgraph of G. This addition is implicit in P but is a crucial addition needed for Q if the lemma is to be true. (Q ⇒ P ): Suppose T is a spanning connected subgraph of G with n − 1 edg...

The maximum planarization problem asks to find a spanning planar subgraph having the largest number of edges. In this paper, we propose two simple selfstabilizing algorithms for this problem in complete bipartite graphs. The first one is an approximation algorithm; it finds a planar subgraph of approximation ratio 0.6 in the average case. Based on the techniques used in the first algorithm, the...

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V (G), dH(x, y) ≤ dG(x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Qn has at least c log2 n · 2n edges.