The rst-order theory of an automatic structure is known to be decidable but there are examples of automatic structures with nonelementary rst-order theories. We prove that the rst-order theory of an automatic structure of bounded degree (meaning that the corresponding Gaifman-graph has

-length incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms

Karger, Motwani and Ramkumar have shown that there is no constant approx-imation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured this is the case even for graphs with bounded degree. On the other hand,Feder, Motwani and Subi have shown that there is a polynomial time

Theorem about individual variable values in LP solution Technique: Solve LP Round some variables

Abstract. We consider testing graph expansion in the bounded-degree graph model. Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural algorithm

We investigate adjacency labeling schemes for graphs of bounded degree ∆ = O(1). In particular, we present an optimal (up to an additive constant) log n+ O(1) adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar

f Garey, Johnson, and Tarjan [GJT76] (it is N P -complete to decide, if a 3-connected 3-regular planar graph has a Hamiltonian path/cycle), and Yannakakis [Yan81] (it is N P -complete to decide, if a graph has a connected spanning subgraph of maximum degree r, r 2). To our knowledge

Abstract. We show that there exists a matching with 4m 5k+3 edges in a graph of degree k and m edges.

In this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter ∆. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set

shown that if k # 2, G is connected and # x#I deg(x) # n - 1 whenever I is a k-element independent set, then G has a spanning tree T with #(T) # k. Here we are interested in the structure of spanning trees under the additional assumption that G does not have a spanning tree with maximum degree less than