Exact Algorithms for Intervalizing Coloured Graphs

Abstract

In the INTERVALIZING COLOURED GRAPHS problem, one must decide for a given graph G = (V , E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses 2 O(n/ log n) time. We also give an O * (2 n ) algorithm for the case that the number of colors is not fixed.

Improved Lower Bounds for Graph Embedding Problems

“…For any fixed k, Intervalizing k-Coloured Graphs can be solved in time 2 O(n/ log n) [4]. Bodlaender and Nederlof [5] conjecture a lower bound (under the Exponential Time Hypothesis) of 2 Ω(n/ log n) time for k ≥ 6; we settle this conjecture and show that it in fact holds for k ≥ 5, even when restricted to trees.…”

“…To complement this result for a bounded number of colours, we also show a 2 Ω(n) -time lower bound for graphs with an unbounded number of colors, assuming the ETH. Note that this result implies that the algorithm from [4] is optimal. A complication in the proof is that to obtain the stated bound, one can only use (on average) a constant number of vertices of each colour (when using O(n) colours).…”

“…As illustrated by the following theorem, these lower bounds are tight. The algorithm is a simpler example of the techniques used in [3,4,5]. There, the authors use isomorphism tests on graphs, here, we use equality of strings.…”