Deciding the Carathéodory number in cycle convexity is NP-complete even on bipartite graphs, with exact values or constant upper bounds obtained for forests, cycles, complete graphs, split graphs, and several other classes.
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The cycle-convexity exchange number is NP-hard to compute in general but has polynomial-time algorithms for chained chordal graphs and explicit formulas for strong and lexicographic products.
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Carath\'eodory Number in Cycle Convexity
Deciding the Carathéodory number in cycle convexity is NP-complete even on bipartite graphs, with exact values or constant upper bounds obtained for forests, cycles, complete graphs, split graphs, and several other classes.
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Computing the Exchange Number in Graphs with respect to Cycle Convexity
The cycle-convexity exchange number is NP-hard to compute in general but has polynomial-time algorithms for chained chordal graphs and explicit formulas for strong and lexicographic products.