Edge Multiway Cut and Node Multiway Cut are NP-complete on subcubic graphs

TL;DR

It is shown that Edge Multiway Cut and Node Multiway Cut are NP-complete on graphs of maximum degree $3$ (also known as subcubic graphs), which improves on a previous degree bound of $11$.

Abstract

We show that Edge Multiway Cut (also called Multiterminal Cut) and Node Multiway Cut are NP-complete on graphs of maximum degree $3$ (also known as subcubic graphs). This improves on a previous degree bound of $11$. Our NP-completeness result holds even for subcubic graphs that are planar.

Figures (3)

• Figure 1: The three different types of multiway cuts that we consider in our paper. In all figures, the red square nodes form the terminal set $T$. In the top left figure, the green lines form an edge multiway cut. In the top right, the green encircled vertices form a node multiway cut not containing a vertex of $T$. In the bottom figure, the green encircled vertices form a node multiway cut that contains two vertices of $T$. The coloured parts depict the components formed after removing the edges/vertices of the multiway cut.
• Figure 2: An example of a graph, namely $P_1+P_5+P_7+S_{2,3,4}$, that belongs to the set ${\cal S}$.
• Figure 16: The figure shows the construction in Theorem \ref{['thm:NMwC:C2']}. The leftmost figure is an instance of Edge Multiway Cut on planar subcubic graphs. The figure in between shows a $2$-subdivision of the instance. The rightmost figure shows the line graph of the subdivided graphs drawn in green. In each figure, the terminals are shown as red squares.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 5
• Corollary 6
• Theorem 6
• Proposition 13
• Lemma 13
• Theorem 13