By Webber R.E.

During this thesis, a number of facets of quadtree representations are analyzed. The quadtree is a hierarchical variable-resolution info constitution compatible for representing the geometric items of special effects, the polygonal maps of laptop cartography, and the digitized photographs of desktop vision.The research of quadtrees is gifted in 3 parts:A) a proper semantics for quadtree algorithms,B) stronger algorithms for manipulating the normal sector quadtree, andC) diversifications of the quadtree technique to the duty of representing polygonal maps.

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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.