Complexity results on locally-balanced $2$-partitions of graphs

TL;DR

The problem of the existence of locally-balanced 2-partition with an open (closed) neighborhood is NP-complete for some restricted classes of graphs and the problem of deciding if a given graph has a locally-balanced 2-partition with an open neighborhood is NP-complete for biregular bipartite graphs and even bipartite graphs with maximum degree 4.

Abstract

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1.$$ A $2$-partition $f^{\prime}$ of a graph $G$ is a \emph{locally-balanced with a closed neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=0\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=1\}\vert \right\vert\leq 1.$$ In this paper we prove that the problem of the existence of locally-balanced $2$-partition with an open (closed) neighborhood is $NP$-complete for some restricted classes of graphs. In particular, we show that the problem of deciding if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete for biregular bipartite graphs and even bipartite graphs with maximum degree $4$, and the problem of deciding if a given graph has a locally-balanced $2$-partition with a closed neighborhood is $NP$-complete even for subcubic bipartite graphs and odd graphs with maximum degree $3$. Last results prove a conjecture of Balikyan and Kamalian.

Figures (4)

• Figure 1: The graph $F_1^i$.
• Figure 2: The graph $F_2$.
• Figure 3: The graph $F_3^i$.
• Figure 4: The graph $F_4^{i_1,l_1,i_2,l_2,i_3,l_3}$.

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4
• Theorem 3.5
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3