Bypassing Erd\H{o}s' Girth Conjecture: Hybrid Stretch and Sourcewise Spanners

An $(\alpha ,\beta )$-spanner of an $n$-vertex graph $G=(V,E)$ is a subgraph $H$ of $G$ satisfying that $dist(u, v, H) \leq \alpha \cdot dist(u, v, G)+\beta$foreverypair$(u, v)\in V \times V$,where$dist(u,v,G’)$ denotes the distance between $u$ and $v$ in $G^{\mathrm{\prime }}\subseteq G$. It is known that for every integer $k\ge 1$, every graph $G$ has a polynomially constructible $(2k-1,0)$-spanner of size $O(n^{1+1\mathrm{/}k})$. This size-stretch bound is essentially optimal by the girth conjecture. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of $2k-1$ only for \emph{neighboring} vertex pairs, while maintaining a strictly \emph{better} multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of \emph{$k$-hybrid spanners}, in which non neighboring vertex pairs enjoy a \emph{multiplicative} $k$-stretch and the neighboring vertex pairs enjoy a \emph{multiplicative} $(2k-1)$ stretch (hence, tight by the conjecture). We show that for every unweighted $n$-vertex graph $G$ with $m$ edges, there is a (polynomially constructible) $k$-hybrid spanner with $O(k^2\cdot n^{1+1\mathrm{/}k})$ edges. \indent An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs $S\times V$ for a given subset of vertices $S\subseteq V$ referred to here as \emph{sources}. Spanners in which the distances in $S\times V$ are bounded are referred to as \emph{sourcewise spanners}. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more).