The Randic index and the diameter of graphs

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is connected. Aouchiche-Hansen-Zheng conjectured that among all connected graphs $G$ on $n$ vertices the path $P_n$ achieves the minimum values for both $R(G)/D(G)$ and $R(G)- D(G)$. We prove this conjecture completely. In fact, we prove a stronger theorem: If $G$ is a connected graph, then $R(G)-(1/2)D(G)\geq \sqrt{2}-1$, with equality if and only if $G$ is a path with at least three vertices.Comment: 17 pages, accepted by Discrete Mathematic

Variants of Plane Diameter Completion

The {\sc Plane Diameter Completion} problem asks, given a plane graph $G$ and a positive integer $d$, if it is a spanning subgraph of a plane graph $H$ that has diameter at most $d$. We examine two variants of this problem where the input comes with another parameter $k$. In the first variant, called BPDC, $k$ upper bounds the total number of edges to be added and in the second, called BFPDC, $k$ upper bounds the number of additional edges per face. We prove that both problems are {\sf NP}-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when $k=1$ on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in $O(n^{3})+2^{2^{O((kd)^2\log d)}}\cdot n$ steps.Comment: Accepted in IPEC 201