An Efficient Algorithm to Test Forcibly-connectedness of Graphical Degree Sequences

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al\u27s classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage\u27s classic algorithm to enumerate graphical partitions of even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length $n$ and forcibly connected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) almost all zero-free graphical degree sequences of length $n$ are forcibly connected; (2) almost none of the graphical partitions of even $n$ are forcibly connected

An efficient algorithm to test forcibly-connectedness of graphical degree sequences

We present an algorithm to test whether a given graphical degree sequence is forcibly connected or not and prove its correctness. We also outline the extensions of the algorithm to test whether a given graphical degree sequence is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through experimental evaluations that the algorithm is efficient on average, though its worst case run time is probably exponential. We also adapt Ruskey et al's classic algorithm to enumerate zero-free graphical degree sequences of length $n$ and Barnes and Savage's classic algorithm to enumerate graphical partitions of even integer $n$ by incorporating our testing algorithm into theirs and then obtain some enumerative results about forcibly connected graphical degree sequences of given length $n$ and forcibly connected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) almost all zero-free graphical degree sequences of length $n$ are forcibly connected; (2) almost none of the graphical partitions of even $n$ are forcibly connected.Comment: 20 pages, 11 table

Curve network interpolation by C1C^1 quadratic B-spline surfaces

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate $C^1$ quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.Comment: With respect to the previous version, this version of the paper is improved. The results have been reorganized and it is more general since it deals with non uniform knot partitions. Accepted for publication in Computer Aided Geometric Design, October 201