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On maximum tolerant Radon partitions for all-paths convexity in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 8، دوره 12، شماره 1، خرداد 2027، صفحه 93-113 اصل مقاله (551.98 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.29418.1986 | ||
| نویسندگان | ||
| Sreekumar Sreedharan1؛ Manoj Changat* 2؛ Kannan Balakrishnan3 | ||
| 1Department of Mathematics, Government College for Women, Thiruvananthapuram-695014, India | ||
| 2Department of Futures Studies, University of Kerala, Thiruvananthapuram-695581, India | ||
| 3Department of Computer Applications, Cochin University of Science and Technology, Kochi-682022, India | ||
| چکیده | ||
| In a connected graph $G$, the all-paths transit function $A(u,v)$, consists of the set of all vertices in the graph $G$ which lies on some path connecting $u$ and $v$. Convexity obtained by the all-paths transit function is called all-paths convexity. A Radon partition of a set $P$ of vertices of a graph $G$ is a partition of $P$ into two disjoint non-empty subsets such that their convex hulls intersect. A Radon partition $(P_t, Q_t)$ of $P$ is called $t$-tolerant Radon partition, if for any set $S\subseteq P$ with $|S|\le t$, the intersection of the convex hulls $\langle P_t\setminus S \rangle \cap \langle Q_t\setminus S \rangle \neq \emptyset$. This paper is devoted to $t$-tolerant Radon partitions for the all-paths convexity of connected simple undirected graphs. It is proved that the minimum number of vertices needed for $t$-tolerant Radon partition is $2t+4$. But, some selection of $2t+4$ vertices of $G$ has a $(t+1)$-tolerant Radon partition. In this paper, we discuss the necessary and sufficient condition to the existence of $(t+1)$-tolerant Radon partition for $2t+4$ vertices of $G$. We also develop algorithms to construct the Radon partition, $t$-tolerant Radon partition, and $(t+1)$-tolerant Radon partition of a set of $2t+4$ vertices, if it exists. | ||
| کلیدواژهها | ||
| All-paths convexity؛ Radon partition؛ $(t+1)$-tolerant Radon partition | ||
| مراجع | ||
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