The \emph{isometric path antichain cover number} of a graph $G$, denoted by $ipacc(G)$, is a graph parameter that was recently introduced to provide a constant factor approximation algorithm for \textsc{Isometric Path Cover}, whose objective is to cover all vertices of a graph with a minimum number of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously shown to be bounded for chordal graphs and, more generally, for graphs of bounded \emph{chordality} and bounded \emph{treelength}. In this paper, we show that the isometric path antichain cover number remains bounded for graphs in three seemingly unrelated graph classes, namely, \emph{hyperbolic graphs}, \emph{(theta, prism, pyramid)-free graphs}, and \emph{outerstring graphs}. Hyperbolic graphs are extensively studied in \emph{Metric Graph Theory}. The class of (theta, prism, pyramid)-free graphs are extensively studied in \emph{Structural Graph Theory}, \textit{e.g.} in the context of the \emph{Strong Perfect Graph Theorem}. The class of outerstring graphs is studied in \emph{Geometric Graph Theory} and \emph{Computational Geometry}. Our results imply a constant factor approximation algorithm for \textsc{Isometric Path Cover} on all the above graph classes. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought.
翻译:{ g$} 的图形 { g$} 的 { memph{ Isology path sublement name} { g$} 是一个图表参数, 最近引入该参数是为了为\ textsc{ Isomatic Path Cover} 提供一个恒定的元素近似算法。 该参数的目标是覆盖具有最小量等度路径( 即, 其尾端之间最短路径 { G$} 的图形 } 。 这个参数先前显示为带框的 { g${ g$} 和( 更广义的) 捆绑定的 \ emph{ choality} 和已绑定的 {emph{ tald} 的图表参数 。 在本文中, 3个看起来无关的图形类别中, 即 \emph{ hypeople { blook } 的图表中, 也显示这些直径直径 。