Vertex-Bipartition: A Unified Approach for Kernelization of Graph Linear Layout Problems Parameterized by Vertex Cover

Abstract

The linear layout of graphs problem asks, given a graph [Formula: see text] and a positive integer [Formula: see text], whether [Formula: see text] admits a layout consisting of a linear ordering of its vertices and a partition of its edges into [Formula: see text] sets such that the edges in each set meet some special requirements. Specific linear layouts include [Formula: see text]-stack layout, [Formula: see text]-queue layout, [Formula: see text]-arch layout, mixed [Formula: see text]-stack [Formula: see text]-queue layout and others. In this paper, we present a unified approach for kernelization of these linear layout problems parameterized by the vertex cover number [Formula: see text] of the input graph. The key point underlying our approach is to partition each set of related vertices into two distinct subsets with respect to the specific layouts, which immediately leads to some efficient reduction rules. We first apply this approach to the mixed [Formula: see text]-stack [Formula: see text]-queue layout problem and show that it admits a kernel of size [Formula: see text], which results in an algorithm running in time [Formula: see text], where [Formula: see text] denotes the size of the input graph. Our work does not only confirm the existence of a fixed-parameter tractable algorithm for this problem mentioned by Bhore et al. (J. Graph Algorithms Appl. 2022), but also derives new results for the [Formula: see text]-stack layout problem and for the [Formula: see text]-queue layout problem respectively. We also employ this approach to the upward [Formula: see text]-stack layout problem and obtain a new result improving that presented by Bhore et al. (GD 2021). Last but not least, we use this approach to the [Formula: see text]-arch layout problem and obtain a similar result.