Independent tree spanners: fault-tolerant spanning trees with constant distance guarantees

AbstractFor any fixed rational parameter t⩾1, a (tree) t-spanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General t-spanners and their variants have multiple applications in the field of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse t-spanners and fault-tolerance by examining independent tree t-spanners. Given a root vertex r, this is a pair of tree t-spanners, such that the two paths from any vertex to r are edge disjoint or internally vertex disjoint, respectively. For t<3, we give a (constructive) linear-time algorithm to decide whether a pair of independent tree t-spanners exist. We also show that the problem for arbitrary t⩾4 in NP-complete. As a less restrictive concept, we also treat tree t-root-spanners, where the distance constraint is relaxed. Here, we show that the problem of deciding the existence of an independent pair of such subgraphs is NP-complete for all non-trivial, rational t. As a special case, we then consider direct tree t-root-spanners. These are tree t-root-spanners where paths from any vertex to the root have to be detour-free. In the edge-independent case, we give a (constructive) linear-time algorithm for deciding the existence of a pair of these for all rational t. The vertex-independent case, however, is shown to be NP-complete

The generalized 3-edge-connectivity of lexicographic product graphs

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. Moreover, all bounds are sharp.Comment: 14 page