Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems

Version 1 : Received: 27 April 2018 / Approved: 2 May 2018 / Online: 2 May 2018 (05:41:54 CEST)

How to cite: Chitturi, B.; Balachander, S.; Satheesh, S.; Puthiyoppil, K. Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems. Preprints 2018, 2018050012 (doi: 10.20944/preprints201805.0012.v1). Chitturi, B.; Balachander, S.; Satheesh, S.; Puthiyoppil, K. Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems. Preprints 2018, 2018050012 (doi: 10.20944/preprints201805.0012.v1).

## Abstract

The independent set, IS, on a graph $G=\left(V,E\right)$ is ${V}^{*}\subseteq V$ such that no two vertices in ${V}^{*}$ have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. ${V}^{*}\subseteq V$ is a vertex cover, i.e. VC of $G=\left(V,E\right)$ if every $e\in E$ is incident upon at least one vertex in ${V}^{*}$ . ${V}^{*}\subseteq V$ is dominating set, DS, of $G=\left(V,E\right)$ if $\forall v\in V$ either $v\in {V}^{*}$ or $\exists u\in {V}^{*}$ and $\left(u,v\right)\in E$ . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if $k=\Theta \left(log\mid V\mid \right)$ then MIS, MVC and MDS can be computed in polynomial time and if $k=O\left({\left(log\mid V\mid \right)}^{\alpha }\right)$ , where $\alpha <1$ , then MCV and MCD can be computed in polynomial time. If $k=\Theta \left({\left(log\mid V\mid \right)}^{1+ϵ}\right)$ , for $ϵ>0$ , then MIS, MVC and MDS require quasi-polynomial time. If $k=\Theta \left(log\mid V\mid \right)$ then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.

## Keywords

NP-complete; graph theory; layered graph; polynomial time; quasi-polynomial time; dynamic programming; independent set; vertex cover; dominating set

## Subject

MATHEMATICS & COMPUTER SCIENCE, General & Theoretical Computer Science

Views 0