Separating path systems for cubic graphs and for complete bipartite graphs
Cristina Fernandes, Carlos Hoppen, George Kontogeorgiou, Guilherme Oliveira Mota, Danni Peng
TL;DR
The paper investigates strongly separating path systems, collections of $P$-paths that separate every pair of edges in a graph. It proves that every connected $2$-degenerate graph on $n$ vertices satisfies $ssp(G) \le n$, via a constructive induction that achieves an $ssp$ with every edge in exactly two paths and every vertex an endpoint of exactly two paths. Building on this, it derives upper bounds for $ssp(G)$ in subcubic, planar, and planar bipartite graphs, and establishes tight bounds for complete bipartite graphs $K_{a,b}$: $ssp(K_{a,b}) = b$ if $a < b/2$, and $ssp(K_{a,b}) \ge (\sqrt{6(b/a)+4}-2)a$ if $b/2 \le a \le b$, with a construction attaining the former in the unbalanced regime. The work combines inductive and decomposition techniques with graceful-labeling-based constructions, yielding near-optimal linear bounds and clarifying the extremes in the bipartite setting. These results advance the understanding of how edge-separation can be achieved efficiently and identify tight regimes in the bipartite case.
Abstract
A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of such a system is the so called strong separation number of $G$. We prove that the strong separation number of every $2$-degenerate graph on $n$ vertices is at most $n$. Using this, we also provide upper bounds for the strong separation number of subcubic graphs, planar graphs, and planar bipartite graphs. On the other hand, we prove that the strong separation number a complete bipartite graph $K_{a,b}$ is at least $b$ if $a<b/2$ and at least $(\sqrt{6(b/2)+4}-2)a$ if $b/2\leq a\leq b$, and we provide a construction that attains the former bound.