Results 1 
6 of
6
GRAPH DRAWINGS WITH FEW SLOPES
, 2006
"... The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a graph with bounded degree. We prove upper and lower bounds on the slopenumber of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slopenumber of interval graphs, cocomparability graphs, and ATfree graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slopenumber at most O(log n). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
On the modelchecking of monadic secondorder formulas with edge set quantifications
, 2010
"... ..."
DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1
An Efficient Partitioning Oracle for BoundedTreewidth Graphs
, 2011
"... Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constanttime algorithms. For any ε> 0, a partitioning oracle provides query access to a fixed partition of the input boundeddegree minorfree graph, in which every component has size poly(1/ε), and the num ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constanttime algorithms. For any ε> 0, a partitioning oracle provides query access to a fixed partition of the input boundeddegree minorfree graph, in which every component has size poly(1/ε), and the number of edges removed is at most εn, where n is the number of vertices in the graph. However, the oracle of Hassidim et al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/ε)) queries to the input graph to answer each query about the partition. Examples of boundedtreewidth graph classes include kouterplanar graphs for fixed k, seriesparallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/ε)time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/ε)time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive εn in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/ε) time whether the input boundedtreewidth graph is kcolorable or perfect. 1
Reduction techniques for Graph Isomorphism in the context of width parameters
 CoRR
"... ar ..."
(Show Context)