An edge of $G$ is singular if it does not lie on any triangle of $G$; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G[N(u)]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle of length at least $4$ with at most $s$ non-singular edges and with at least $s-5$ locally connected vertices; (ii) for each locally disconnected vertex $v$ of degree $2$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle $C$ with at most $s$ non-singular edges and with at least $s-3$ locally connected vertices and such that $G[V (C)\cap V_{2} (G)]$ is a path or a cycle, then $G$ has a 2-factor, and it is the best possible in some sense. This result generalizes two known results in Faudree, Faudree and Ryjáček (2008) and in Ryjáček, Xiong and Yoshimoto (2010).
This paper proposes a non-trivial definition of the notion of analytic method. Working within the so-called instructional model of method, I distinguish three kinds of instructions which occur in methods: selective, executive, and declarative instructions. I discuss the relation between each of these and the analyticity of a method. Then I define the notions of an analytic use of an instruction and of an analytic instruction, which are at the basis of the proposed definition of an analytic method. Finally, I discuss the issue of circularity in the presented model which arises if we consider a finite agent testing a method for analyticity., Tato práce navrhuje netriviální definici pojmu analytická metoda. V rámci tzv. Instruktážního modelu metody rozlišuji tři druhy instrukcí, které se vyskytují v metodách: selektivní, exekutivní a deklarativní . Diskutuji o vztahu mezi každou z nich a analytičnosti metody. Dále definuji pojmy analytického použití instrukce a analytické instrukce , které jsou základem navrhované definice analytické metody. Závěrem se zabývám otázkou kruhovitosti v prezentovaném modelu, která vzniká, pokud uvažujeme konečný agent testující metodu analyticity., and Miloš Kosterec