Navigating in a Graph by Aid of Its Spanning Tree Metric

• Authors: Feodor F. Dragan;Martin Matamala
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2011

Let $G=(V,E)$ be a graph and $T$ be a spanning tree of $G$. We consider the following strategy in advancing in $G$ from a vertex $x$ towards a target vertex $y$: from a current vertex $z$ (initially, $z=x$), unless $z=y$, go to a neighbor of $z$ in $G$ that is closest to $y$ in $T$ (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in $G$ and can use the distances in $T$ to navigate in $G$. Thus, additionally to standard local information (the neighborhood $N_G(v)$), the only global information that is available to each vertex $v$ is the topology of the spanning tree $T$ (in fact, $v$ can know only a very small piece of information about $T$ and still be able to infer from it the necessary tree-distances). For each source vertex $x$ and target vertex $y$, this way, a path, called a greedy routing path, is produced. Denote by $g_{G,T}(x,y)$ the length of a longest greedy routing path that can be produced for $x$ and $y$ using this strategy and $T$. We say that a spanning tree $T$ of a graph $G$ is an additive $r$-carcass for $G$ if $g_{G,T}(x,y)\leq d_G(x,y)+r$ for each ordered pair $x,y\in V$. In this paper, we investigate the problem, given a graph family $\mathcal{F}$, of whether a small integer $r$ exists such that any graph $G\in\mathcal{F}$ admits an additive $r$-carcass. We show that rectilinear $p\times q$ grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs) all admit additive 0-carcasses. Furthermore, every chordal graph $G$ admits an additive $(\omega+1)$-carcass (where $\omega$ is the size of a maximum clique of $G$), each 3-sun-free chordal graph admits an additive 2-carcass, and each chordal bipartite graph admits an additive 4-carcass. In particular, any $k$-tree admits an additive $(k+2)$-carcass. All those carcasses are easy to construct in sequential as well as in distributed settings.