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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.
Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general position set i
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
In this paper, based on the contributions of Tucker (1983) and Seb{H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable forcing structure
Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as
For nonnegative integers $k, d_1, ldots, d_k$, a graph is $(d_1, ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $iin{1, ldots, k}$. A class $