Completely Independent Spanning Trees in Line Graphs

Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2} \rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely independent spanning trees still exist in the graph obtained from $L(K_n)$ by deleting any vertex (respectively, any induced path of order at most $\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where $\delta(G)$ denotes the minimum degree of $G$. $\$ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is not super edge-connected or $\delta(G) \geq 2k$. $\$ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$ has $k$ completely independent spanning trees if $G$ is regular. $\$ $\bullet$ Every $(k^2+2k-1)$-connected line graph $L(G)$ with $\delta(G) \geq k+1$ has $k$ completely independent spanning trees.Comment: 20 pages with 5 figure

On Minimum Average Stretch Spanning Trees in Polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.Comment: 17 pages, 12 figure

Orderly Spanning Trees with Applications

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51