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Input: A graph G and a starting vertex root of G. Output: Goal state.The parent links trace the shortest path back to root [9]. 1 procedure BFS(G, root) is 2 let Q be a queue 3 label root as explored 4 Q.enqueue(root) 5 while Q is not empty do 6 v := Q.dequeue() 7 if v is the goal then 8 return v 9 for all edges from v to w in G.adjacentEdges(v) do 10 if w is not labeled as explored then 11 ...
The width of a node is the number of its parents, and the width of an ordered graph is the maximal width of its nodes. The induced graph of an ordered graph is obtained by adding some edges to an ordering graph, using the method outlined below. The induced width of an ordered graph is the width of its induced graph. [2] Given an ordered graph ...
A graph of cutwidth 2. For the left-to-right vertex ordering shown, each vertical line crosses at most two edges. In graph theory, the cutwidth of an undirected graph is the smallest integer with the following property: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at ...
The largest component has logarithmic size. The graph is a pseudoforest. Most of its components are trees: the number of vertices in components that have cycles grows more slowly than any unbounded function of the number of vertices. Every tree of fixed size occurs linearly many times. [29] Critical /
Parameterized complexity is the complexity-theoretic study of problems that are naturally equipped with a small integer parameter k and for which the problem becomes more difficult as k increases, such as finding k-cliques in graphs. A problem is said to be fixed-parameter tractable if there is an algorithm for solving it on inputs of size n ...
Construction of a distance-hereditary graph of clique-width 3 by disjoint unions, relabelings, and label-joins. Vertex labels are shown as colors. In graph theory, the clique-width of a graph G is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs.
The graph structure theorem for minor-closed graph families states that, for any such family F, the graphs in F can be decomposed into clique-sums of graphs that can be embedded onto surfaces of bounded genus, together with a bounded number of apexes and vortices for each component of the clique-sum. An apex is a vertex that may be adjacent to ...
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [4] [5] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [6] or they can be used to build a pointer based quadtree.