Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Packing 3-vertex paths in claw-free graphs and related topics

An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G - {x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E is a set of three edges in G, then G - E has an L-factor if and only if the subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1 mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G, (a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3. We explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings. We also discuss relations between L-packing and domination problems as well as between induced L-packings and the Hadwiger conjecture. Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page

Perfect Matchings in Claw-free Cubic Graphs

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure

On strong rainbow connection number

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$ path of length $d(u,v)$, where $d(u,v)$ is the distance between $u$ and $v$. The graph $G$ is strongly rainbow connected if there exists a rainbow $u-v$ geodesic for any two vertices $u$ and $v$ in $G$. The strong rainbow connection number of $G$, denoted $src(G)$, is the minimum number of colors that are needed in order to make $G$ strong rainbow connected. In this paper, we first investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that $G$ is a tree if and only if $src(G)=m$, we will show that $src(G)\neq m-1$, so $G$ is not a tree if and only if $src(G)\leq m-2$, where $m$ is the number of edge of $G$. Furthermore, we characterize the graphs $G$ with $src(G)=m-2$. We next give a sharp upper bound for $src(G)$ according to the number of edge-disjoint triangles in graph $G$, and give a necessary and sufficient condition for the equality.Comment: 16 page