By Richard J. Sadus, Sadus

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For each vertex x we introduce a pointer current(x), indicating the current edge in the list containing all edges in δ(x) or δ + (x) (this list is part of the input). Initially current(x) is set to the ﬁrst element of the list. In 3 , the pointer moves forward. When the end of 26 2. Graphs the list is reached, x is removed from Q and will never be inserted again. e. O(n + m). To identify the connected components of a graph, we apply the algorithm once and check if R = V (G). If so, the graph is connected.

There exists a vertex z such that G − {x, y, z} is disconnected. Since {v, w} ∈ E(G), there exists a connected component D of G − {x, y, z} which contains neither v nor w. But D contains a neighbour d of y, since otherwise D is a connected component of G − {x, z} (again contradicting the fact that G is 3-connected). So d ∈ V (D) ∩ V (C), and thus D is a subgraph of C. Since y ∈ V (C) \ V (D), we have a contradiction to the minimality of |V (C)|. 37. (Kuratowski [1930], Wagner [1937]) A 3-connected graph is planar if and only if it contains neither K 5 nor K 3,3 as a minor.

Let r ∈ U . As noted above, F := {X ∈ F : r ∈ X } ∪ {U \ X : X ∈ F, r ∈ X } is laminar, so let (T, ϕ) be a tree-representation of (U, F ). Now for an edge e ∈ E(T ) there are three cases: If Se ∈ F and U \ Se ∈ F, we replace the edge e = (x, y) by two edges (x, z) and (y, z), where z is a new vertex. If Se ∈ F and U \ Se ∈ F, we replace the edge e = (x, y) by (y, x). If Se ∈ F and U \ Se ∈ F, we do nothing. Let T be the resulting graph. Then (T , ϕ) is a tree-representation of (U, F). ✷ The above result is mentioned by Edmonds and Giles [1977] but was probably known earlier.