By considering the order of the largest induced bipartite subgraph of \(G\), Hagauer and Klaviar [4] were able to improve the bounds first published by V. G. Vizing [6] for the independence number of the Cartesian product \(G \Box H\) for any graph \(H\). In this paper, we study maximum independent sets in \(G \Box H\) when \(G\) is a caterpillar, and derive bounds for the independence number when \(H\) is bipartite. The upper bound we produce is less than or equal to that in [4] when \(H\) is also a caterpillar, and is shown to be strictly smaller when \(H\) comes from a restricted class of caterpillars.
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