Backward Arcs in Hamilton Oriented Cycles and Paths in Directed Graphs with Independence Number Two
This provides theoretical guarantees for graph connectivity in combinatorial optimization, but it is incremental as it builds on prior work by Freschi and Lo.
The paper tackles the problem of finding Hamilton oriented cycles and paths in directed graphs with independence number at most two, proving that every 2-connected digraph has a Hamilton oriented cycle with at most five backward arcs and every 1-connected digraph has a Hamilton oriented path with at most two backward arcs.
In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a forward arc of $P$; otherwise, $x_{i+1}x_{i}$ is a backward arc. The independence number $α(D)$ is the maximum integer $p$ such that $D$ has a set of $p$ vertices where there is no arc between any pair of vertices. A digraph is $k$-connected if its underlying undirected graph is $k$-connected. Freschi and Lo (JCT-B 2024) proved that every $n$-vertex oriented graph with minimum degree $δ\ge n/2$ has a Hamilton oriented cycle with at most $n-δ$ backward arcs. We prove that every 2-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented cycle with at most five backward arcs, and every 1-connected digraph $D$ with $α(D)\le 2$ has a Hamilton oriented path with at most two backward arcs.