Additive Tree Spanners

Graph-Theoretic Concepts in Computer Science

Köhler

et al. 2005

Abstract. In this paper we study the existence of a small set T of spanning trees that collectively "1-span" an interval graph G. In particular, for any pair of vertices u, v we require a tree T ∈ T such that the distance between u and v in T is at most one more than their distance in G. We show that:-there is no constant size set of collective tree 1-spanners for interval graphs (even unit interval graphs), -interval graph G has a set of collective tree 1-spanners of size O(log D), where D is the diameter of G, -interval graphs have a 1-spanner with fewer than 2n − 2 edges. Furthermore, at the end of the paper we state other results on collective tree c-spanners for c > 1 and other more general graph classes.

Section: Lower Boundsupporting

confidence: 81%

“…(Recall that interval graphs have a single tree that 2-spans the graph [11,8].) In the journal version of the paper, we will present proofs of the following theorems.…”

Section: Discussionmentioning

confidence: 99%

Collective Tree 1-Spanners for Interval Graphs

Corneil

Graph-Theoretic Concepts in Computer Science

Köhler

et al. 2005

“…It is known that the class of chordal graphs does not admit any good tree spanners. Independently McKee [21] and Kratsch et al [16] showed that, for every fixed integer t, there is a chordal graph without tree t-spanners (additive as well as multiplicative). Furthermore, recently Brandstädt et al [5] have shown that, for any t ≥ 4, the problem to decide whether a given chordal graph G admits a multiplicative tree t-spanner is NP-complete.…”

Section: Previous Results and Their Implicationsmentioning

confidence: 99%

Distance Approximating Trees: Complexity and Algorithms

Lecture Notes in Computer Science

Yan

2006

Abstract. Let Δ ≥ 1 and δ ≥ 0 be real numbers. A treeThe distance (Δ, δ)-approximating tree problem asks for a given graph G to decide whether G has a distance (Δ, δ)-approximating tree. In this paper, we consider unweighted graphs and show that the distance (Δ, 0)-approximating tree problem is NP-complete for any Δ ≥ 5 and the distance (1, 1)-approximating tree problem is polynomial time solvable.

“…Recall that 2-trees do not have any tree r-spanners (additive or multiplicative) with a constant r (see, e.g., [29]). We also have the following result.…”

Section: Theorem 4 Any Shortest Path Tree Of a Chordal Graph G Is Anmentioning

confidence: 99%

Navigating in a Graph by Aid of Its Spanning Tree

Algorithms and Computation

Matamala

2008

Abstract. Let G = (V, E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood NG(v)), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by gG,T (x, y) the length of a longest greedy routing path that can be produced for x and y using this strategy and T . We say that a spanning tree T of a graph G is an additive r-carcass for G if gG,T (x, y) ≤ dG(x, y) + r for each ordered pair x, y ∈ V . In this paper, we investigate the problem, given a graph family F, whether a small integer r exists such that any graph G ∈ F admits an additive r-carcass. We show that rectilinear p × q grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive (ω(G) + 1)-carcass (where ω(G) is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive (k +2)-carcass. All those carcasses are easy to construct.