On Distance Preserving and Sequentially Distance Preserving Graphs
Jason P. Smith, Emad Zahedi
TL;DR
The paper studies when graphs preserve pairwise distances under isometric subgraphs (dp) and the stronger sequential variant (sdp). It connects sdp to $k$-simplicial orderings, proving that all $4$-chordal graphs are sdp and thus dp, and it provides a decomposition framework for dp in separable graphs of the form $G+_x H$ with a cut vertex $x$, including a formula for $\mathrm{dp}(G+_x H)$. It further examines non-dp graphs by introducing the cycle-based family $\mathcal{C}_{k,\ell}$ and establishing a criterion $k>2(\ell+2)$ under which these graphs remain non-dp, offering a constructive route to generate non-dp graphs from cycles. Collectively, these results clarify structural boundaries between dp and non-dp graphs, yield testable tools for dp in composite graphs, and suggest open directions for broader characterization and algorithmic development within metric graph theory.
Abstract
A graph $H$ is an \emph{isometric} subgraph of $G$ if $d_H(u,v)= d_G(u,v)$, for every pair~$u,v\in V(H)$. A graph is \emph{distance preserving} if it has an isometric subgraph of every possible order. A graph is \emph{sequentially distance preserving} if its vertices can be ordered such that deleting the first $i$ vertices results in an isometric subgraph, for all $i\ge1$. We give an equivalent condition to sequentially distance preserving based upon simplicial orderings. Using this condition, we prove that if a graph does not contain any induced cycles of length~$5$ or greater, then it is sequentially distance preserving and thus distance preserving. Next we consider the distance preserving property on graphs with a cut vertex. Finally, we define a family of non-distance preserving graphs constructed from cycles.