... degree of xiis 2, and the degree of a0is ∆ − 1. However, the degree of a1isthesameinbothtrees,soS wins either by Rule 2 or by Rule 3. Since P is a maximum path in G, no point of V −P ... sum of the degrees of the vertices insuch a set is exactly n − 1. However, when three or more of the cliques have two or morepoints, the graph does not have a spanning caterpillar of maximum degree ... proof of the claim. Claim 3. Let i ∈{1, 2, ,r}.Ifq>0thenxia1∈ E.Proof. Suppose that q>0andxia1∈ E.FormS by removing the edge a0a1and addingthe edge xia1.InS, the degree...