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تسجيل مستخدم جديدOn monophonic position sets in graphs
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The general position number of a graph $G$ is the size of the largest set $K$ of vertices of $G$ such that no shortest path of $G$ contains three vertices of $K$. In this paper we discuss a related invariant, the monophonic position number, which is obtained from the definition of general position number by replacing `shortest path with `induced path. We prove some basic properties of this invariant and determine the monophonic position number of several common types of graphs, including block graphs, unicyclic graphs, complements of bipartite graphs and split graphs. We present an upper bound for the monophonic position numbers of triangle-free graphs and use it to determine the monophonic position numbers of the Petersen graph and the Heawood graph. We then determine realisation results for the general position number, monophonic position number and monophonic hull number and finally find the monophonic position numbers of joins and corona products of graphs.
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A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${rm gp}(G)$ of $G$.
It is proved that $Ssubseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${rm diam}(G) = 2$, then ${rm gp}(G)$ is the maximum of $omega(G)$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, ${rm gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then ${rm gp}(G) leq alpha(G)$ with equality if ${rm diam}(G) in {2,3}$. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path in p
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