# Layered graphs: a class that admits polynomial time solutions for some hard problems

## TL;DR

This paper introduces layered graphs, a new class where certain NP-hard problems like MIS, MVC, CVC, MDS, and CDS can be solved in polynomial time when each layer has logarithmic size, expanding the scope of efficiently solvable graph problems.

## Contribution

The paper defines layered graphs and proves polynomial-time algorithms for classic NP-hard problems on this class, under specific size constraints for layers.

## Key findings

- Polynomial-time algorithms for MIS, MVC, CVC, MDS, and CDS on layered graphs.
- Layered graphs with layers of size O(log |V|) generalize known polynomial cases.
- Restrictions like bipartiteness or planarity are not required for these results.

## Abstract

The independent set on a graph $G=(V,E)$ is a subset of $V$ such that no two vertices in the subset have an edge between them. The MIS problem on $G$ seeks to identify an independent set with maximum cardinality, i.e. maximum independent set or MIS. $V* \subseteq V$ is a vertex cover $G=(V,E)$ if every edge in the graph is incident upon at least one vertex in $V*$. $V* \subseteq V$ is dominating set of $G=(V,E)$ if forall $v \in V$ either $v \in V*$ or $\exists u \in V*$ and $(u,v) \in E$. A connected dominating set, CDS, is a dominating set that forms a single component in $G$. The MVC problem on $G$ seeks to identify a vertex cover with minimum cardinality, i.e. minimum vertex cover or MVC. Likewise, CVC seeks a connected vertex cover (CVC) with minimum cardinality. The problems MDS and CDS seek to identify a dominating set and a connected dominating set respectively of minimum cardinalities. MVC, CVC, MDS, and CDS on a general graph are known to be NP-complete. On certain classes of graphs they can be computed in polynomial time. Such 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. In this article we introduce a new class of graphs called a layered graph and show that if the number of vertices in a layer is $O(\log \mid V \mid)$ then MIS, MVC, CVC, MDS and CDC can be computed in polynomial time. The restrictions that are employed on graph classes that admit polynomial time solutions for hard problems, e.g. lack of cycles, bipartiteness, planarity etc. are not applicable for this class. \\ Key words: Independent set, vertex cover, dominating set, dynamic programming, complexity, polynomial time algorithms.