Additive Spanners for Circle Graphs and Polygonal Graphs

Theoretical Computer Science, 2014

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph G admits a system of µ collective additive tree r-spanners (resp., multiplicative tree t-spanners) if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r (resp., dT (x, y) ≤ t · dG(x, y)). When µ = 1 one gets the notion of additive tree r-spanner (resp., multiplicative tree t-spanner). It is known that if a graph G has a multiplicative tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G. We use this to demonstrate that there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative tree t-spanner, constructs a system of at most log 2 n collective additive tree O(t log n)-spanners of G. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph G admits a multiplicative t-spanner with tree-width k − 1, then G admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most k disks of G of radius at most ⌈t/2⌉ each. This is used to demonstrate that, for every fixed k, there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative t-spanner with tree-width k − 1, constructs a system of at most k(1 + log 2 n) collective additive tree O(t log n)-spanners of G. a stretch t [17], and an additive tree r-spanner of G is a spanning tree with a surplus r [59]. If we approximate the graph by a tree spanner, we can solve the problem on the tree and the solution interpret on the original graph. The tree t-spanner problem asks, given a graph G and a positive number t, whether G admits a tree t-spanner. Note that the problem of finding a tree t-spanner of G minimizing t is known in literature also as the Minimum Max-Stretch spanning Tree problem (see, e.g., and literature cited therein).

SIAM Journal on Discrete Mathematics, 2006

In this paper we introduce a new notion of collective tree spanners. We say that a graph G = (V, E) admits a system of µ collective additive tree r-spanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2-spanners and any c-chordal graph admits a system of at most log 2 n collective additive tree (2 c/2)-spanners. Towards establishing these results, we present a general property for graphs, called (α, r)decomposition, and show that any (α, r)-decomposable graph G with n vertices admits a system of at most log 1/α n collective additive tree 2rspanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.

Lecture Notes in Computer Science, 2008

In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distance-hereditary graphs and to show that the Steiner tree problem can still be solved in polynomial time on this more general class of graphs. In this paper, we demonstrate that every n-vertex homogeneously orderable graph G admits a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤ dG(x, y) + 3 (i.e., an additive tree 3-spanner) and a system T (G) of at most O(log n) spanning trees such that, for any two vertices x, y of G, a spanning tree T ∈ T (G) exists with dT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collective additive tree 2-spanners). These results generalize known results on tree spanners of dually chordal graphs and of distance-hereditary graphs. The results above are also complemented with some lower bounds which say that on some n-vertex homogeneously orderable graphs any system of collective additive tree 1-spanners must have at least Ω(n) spanning trees and there is no system of collective additive tree 2-spanners with constant number of trees.

Algorithmica, 2010

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G = (V , E) admits a system of μ collective additive tree r-spanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that d T (x, y) ≤ d G (x, y) + r. We describe a general method for constructing a "small" system of collective additive tree r-spanners with small values of r for "well" decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O( √ n) collective additive tree 0-spanners, any weighted graph with tree-width at most k − 1 admits a system of k log 2 n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of k log 3/2 n collective additive tree (2w)-spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2 c/2 w)-spanners and a system of 4 log 2 n collective additive tree (2( c/3 + 1)w)spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log 2 n collective additive tree (2w)-spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.

Journal of Computer and System Sciences, 2011

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the Tree t-Spanner problem (the problem to decide whether G admits a tree t-spanner) is polynomial time solvable for t ≤ 3 and is NP-complete as long as t is part of the input. They also left as an open problem whether the Tree t-Spanner problem is polynomial time solvable for every fixed t ≥ 4. In this work we resolve this open problem and extend the solution in several directions. We show that for every fixed t, it is possible in polynomial time not only to decide if a planar graph G has a tree t-spanner, but also to decide if G has a t-spanner of bounded treewidth. Moreover, for every fixed values of t and k, the problem, for a given planar graph G to decide if G has a t-spanner of treewidth at most k, is not only polynomial time solvable, but is fixed parameter tractable (with k and t being the parameters). In particular, the running time of our algorithm is linear with respect to the size of G. We extend this result from planar to a much more general class of sparse graphs containing graphs of bounded genus. An apex graph is a graph obtained from a planar graph G by adding a vertex and making it adjacent to some vertices of G. We show that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. We prove that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs. In particular, for every t ≥ 4, the problem of finding a tree t-spanner is NP-complete on K 6 -minor-free graphs. Thus our results are tight, in a sense that the restriction of input graph being apex-minor-free cannot be replaced by H-minor-free for some non-apex fixed graph H. Graphs of bounded treewidth are sparse graphs and our technique can be used to settle the complexity of the parameterized version of the Sparsest t-Spanner problem, where for given t and m one asks if a given n-vertex graph has a t-spanner with at most n − 1 + m edges. Our results imply that the Sparsest t-Spanner problem is fixed parameter tractable on apex-minor-free graphs with t and m being the parameters. Finally, we show that the optimization version of the Sparsest t-Spanner problem, which asks for a t-spanner with the minimum number of edges, admits PTAS for apexminor-free graphs. This resolves an open question asked by Duckworth, Wormald, and Zito. * A preliminary version of these results appeared in the proceedings of the 35th International Colloquium PROBLEM: k-Treewidth t-spanner INSTANCE: A connected graph G and integers k and t. QUESTION: Is there a t-spanner S of G of treewidth at most k?

This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most , i.e., the tree-length graphs. For such graphs we construct additive 2 -spanners with O( n+n logn) edges, and additive 4 -spanners with O( n) edges. This provides new upper bounds for chordal graphs for which = 1. We also show a lower bound, and prove that there are graphs of tree-length for which every multiplicative -spanner (and thus every additive ( 1)-spanner) requires (n1+1/ () ) edges. c 2007 Elsevier B.V. All rights reserved.

Lecture Notes in Computer Science, 2003

In this paper we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n−2 edges and an additive 3-spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(1989), 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m · log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n − 2 edges which is constructable in O(n · k + m) time.

Theoretical Computer Science, 2007

This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most δ, i.e., the tree-length δ graphs. For such graphs we construct additive 2δ-spanners with O(δn+n log n) edges, and additive 4δ-spanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) requires Ω (n 1+1/Θ(δ) ) edges.

Theoretical Computer Science, 2005

In this paper, we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n − 2 edges and an additive 3-spanner with at most O(n log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory 13 (1989) 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n − 2 edges which is constructable in O(n k + m) time.

Theoretical Computer Science, 2004

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The TREE t-SPANNER problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359 -387) by showing that, for any t ¿ 4, TREE t-SPANNER is NP-complete even on chordal graphs of diameter at most t + 1 (if t is even), respectively, at most t + 2 (if t is odd). Then we point out that every chordal graph of diameter at most t − 1 (respectively, t − 2) admits a tree t-spanner whenever t ¿ 2 is even (respectively, t ¿ 3 is odd), and such a tree spanner can be constructed in linear time.