Preprint Article Version 1 This version is not peer-reviewed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 = ( V , E ) is V * 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 * V is a vertex cover, i.e. VC of G = ( V , E ) if every e E is incident upon at least one vertex in V * . V * V is dominating set, DS, of G = ( V , E ) if v V either v V * or u V * and ( u , v ) 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 = Θ ( log V ) then MIS, MVC and MDS can be computed in polynomial time and if k = O ( ( log V ) α ) , where α < 1 , then MCV and MCD can be computed in polynomial time. If k = Θ ( ( log V ) 1 + ϵ ) , for ϵ > 0 , then MIS, MVC and MDS require quasi-polynomial time. If k = Θ ( log V ) then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.

Subject Areas

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

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