Deciding whether a planar graph has a cubic subgraph is NP-complete

AbstractWe show that the problem of deciding whether a given planar graph (complete with planar embedding) of degree at most 7 has a cubic subgraph is NP-complete

Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs

A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms

Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016