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Abstract: Došlić and Måløy (2010) obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the -cactus chains with the most matchings. [Copyright &y& Elsevier]

Abstract: A word is a fixed point of a nontrivial morphism if and is not the identity on the alphabet of . The paper presents the first polynomial algorithm deciding whether a given finite word is such a fixed point. The algorithm also constructs the corresponding morphism, which has the smallest possible number of non-erased letters. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed set with . For and , define by and We call a super-stationary set if holds for any infinite subset of . Denoting the derived set (i.e. the set of accumulating points) of and with , it is known [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subset of such that there exists an infinite subset of with , there exists an infinite subset such that is a super-stationary set. Moreover, if for any infinite subset of , then the maximal pattern complexity [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] is . Thus, the uniform complexity functions are realized by the super-stationary sets [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call a super-subword of if there exists with such that . Let be the set of having no super-subword . Denote where . In this paper, we prove that the class of super-stationary sets other than coincides with the class of for nonempty finite sets . Moreover, it also coincides with the class of for nonempty finite sets , where is the set of minimal covers of . Using these expressions, we can calculate the complexity of super-stationary sets and prove that the complexity function of a super-stationary set in is either or a polynomial function of for large . We also discuss the word problems related to the super-subwords. [Copyright &y& Elsevier]

Abstract: Let be the chromatic polynomial of a graph . A graph is chromatically unique if for any graph , implies H is isomorphic to . For integers , , denote by the complete -partite graph that has partite sets of size and one partite set of size . Let be the set of graphs obtained from by adding a set of edges to the partite set of size such that is bipartite. If , denote the only graph in by . In this paper, we shall prove that for and , each graph is chromatically unique if and only if is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph is chromatically unique for and . [Copyright &y& Elsevier]