The subgraph homeomorphism problem has been shown by Robertson and Seymour to be polynomial-time solvable for any fixed pattern graph H . The result, however, is not practical, involving constants that are worse than exponential in |H |. Practical algorithms have only been developed for a few specific pattern graphs, the most recent of these being the wheels with four and five spokes. This paper looks at the subgraph homeomorphism problem where the pattern graph is a wheel with six spokes. The main result is a theorem characterizing graphs that do not contain subdivisions of W 6 . We give an efficient algorithm for solving the subgraph homeomorphism problem for W 6 . We also give a strengthening of the previous W 5 result.
The Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.
We define a parameter which measures the proportion of vertices which must be removed from any graph in a class in order to break the graph up into small (i.e. bounded sized) components. We call this the coefficient of fragmentability of the class. We establish values or bounds for the coefficient for various classes of graphs, particularly graphs of bounded degree. Our main upper bound is proved by establishing an upper bound on the number of vertices which must be removed from a graph of bounded degree in order to leave a planar graph.
Academic Press
The chromatic polynomial gives the number of proper $\lambda$-colourings of a graph $G$. This paper considers factorisation of the chromatic polynomial as a first step in an algebraic study of the roots of this polynomial. The chromatic polynomial of a graph is said to have a chromatic factorisation if $P({G},\lambda)=P({H_{1}},\lambda)P({H_{2}},\lambda)/P({K_{r}},\lambda)$ for some graphs $H_{1}$ and $H_{2}$ and clique $K_{r}$. It is known that the chromatic polynomial of any clique-separable graph, that is, a graph containing a separating $r$-clique, has a chromatic factorisation. We show that there exist other chromatic polynomials that have chromatic factorisations but are not the chromatic polynomial of any clique-separable graph and identify all such chromatic polynomials of degree at most 10. We introduce the notion of a certificate of factorisation, that is, a sequence of algebraic transformations based on identities for the chromatic polynomial that explains the factorisations for a graph. We find an upper bound of $n^{2}2^{n^{2}/2}$ for the lengths of these certificates, and find much smaller certificates for all chromatic factorisations of graphs of order $\leq 9$.
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