7 research outputs found
Boxicity of Series Parallel Graphs
The three well-known graph classes, planar graphs (P), series-parallel
graphs(SP) and outer planar graphs(OP) satisfy the following proper inclusion
relation: OP C SP C P. It is known that box(G) <= 3 if G belongs to P and
box(G) <= 2 if G belongs to OP. Thus it is interesting to decide whether the
maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In
this paper we construct a series-parallel graph with boxicity 3, thus resolving
this question. Recently Chandran and Sivadasan showed that for any G, box(G) <=
treewidth(G)+2. They conjecture that for any k, there exists a k-tree with
boxicity k+1. (This would show that their upper bound is tight but for an
additive factor of 1, since the treewidth of any k-tree equals k.) The
series-parallel graph we construct in this paper is a 2-tree with boxicity 3
and is thus a first step towards proving their conjecture.Comment: 10 pages, 0 figure
Intersection Dimension and Maximum Degree
We show that the intersection dimension of graphs with respect to several hereditary graph classes can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O(∆ log ∆ log log ∆ ). We also obtain bounds in terms of treewidth