Abstract
The spread of influence in a social network, diseases in a community, or failures on an interconnected site are topics studied in various fields. Convexity in graphs is a powerful framework for modeling this diffusion behavior. The Carathéodory number is an interesting convexity parameter that can be analyzed to understand the dynamics of this behavior. It is known to be NP-Complete. In this paper, we establish some bounds on the P3-Carathéodory number of graphs with diameter two. For a biconnected diameter-two graph G it holds that c(G)≤ ∆ +1, while diameter-two with cut-vertex has c(G)= 2. In addition, we show that the P3-Caratheodory number of biconnected C6-free diameter-two graphs is at most 4.
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