In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairns’s original proof required exponentially many morphing steps. We prove that there is a morph that consists ofO(...

A total dominating set of a graph G=(V,E) is subset D V such that every vertex in adjacent to at least one D. The domination number G, denoted by ?t(G), the minimum cardinality G. near-triangulation biconnected planar admits plane embedding all its faces are triangles except possibly outer face. We show this paper ?t(G)??2n5? for any G order n?5, with two exceptions.

A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S. The problem of deciding whether a plane graph admits a point-set embedding on a given set of points is NPcomplete for 2-connected planar graphs, but polynomial-time solvable for outerplanar graphs and plane 3-trees. In this ...

We study the problem of finding minimum multicuts for an undirected planar graph, where all the terminal vertices are on the boundary of the outer face. This is known as an Okamura-Seymour instance. We show that for such an instance, the minimum multicut problem can be reduced to the minimum-cost Steiner forest problem on a suitably defined dual graph. The minimum-cost Steiner forest problem ha...

An outerplanar graph is a planar graph which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many semi-structured data like the NCI dataset having about 250,000 chemical compounds can be expressed by outerplanar graphs. In this paper, we consider a data mining problem of extracting structural features from semi-structured data like the NCI dataset. For...

We study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2...n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses...

In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform AC0. This provides a striking contrast to the non-planar case, where exactly NC1 is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the oute...

Let On be the set of all maximal outer-planar graphs order n. ar(On,F) denote maximum positive integer k such that there is a k-edge-coloring graph T in family which has no rainbow subgraph F. Denote by Mk matching size k. In this paper, we prove ar(On,Mk)≤n+4k−9 for n≥3k−3, expressively improves existing upper bound ar(On,Mk). We also ar(On,M5)=n+4 n≥15.