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PQ Trees, PC Trees, and Planar Graphs
Wen-Lian Hsu
,
Ross M. McConnell
Published
2004
in
Handbook of Data Structures and Applications
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Abstract
Colorado State University 1.
Topics
14 Figures and Tables
FIGURE 1.13: A C node x of G represents a cycle in H , so a path of G through x represents two possible paths in H .
FIGURE 1.15: If w is a P node, S&H gets a K3,3 with bipartition {{z1, z2, z3}, {w, i, t}}. This K3,3 is the contraction of a K3,3 of the original input graph H .
FIGURE 1.16: If w is a C node but at least one of z1, z2, and z3 is a non-neighbor of w, the case can be reduced to that of Figure 1.15 by selecting a neighbor w′ to serve in the role of w.
FIGURE 1.18: If a C node x has four neighbors (a, b, c, d) in that order, such that there are disjoint paths from a and c to i and b and d to an ancestor of i, then the cycle that x represents, together with these paths, gives a K3,3.
FIGURE 1.2: An interval graph is the intersection graph of a set of intervals on a line. There is one vertex for each of the intervals, and two vertices are adjacent if and only if the corresponding intervals intersect.
FIGURE 1.6: Assigning a new zero column x to a matrix, computing the PC tree for it, and then picking the PC tree up at x to root it, gives the PQ tree for the matrix, rooted at y, when the C nodes are reinterpreted as Q nodes.
FIGURE 1.9: An order-preserving contraction of an edge xy. The neighbors of x and y are cyclically ordered. The edge is removed and x and y are identified, so that the cyclic order of neighbors of x and y about the edge is preserved.
FIGURE 1.13: A C node x of G represents a cycle in H , so a path of G through x represents two possible paths in H .
FIGURE 1.15: If w is a P node, S&H gets a K3,3 with bipartition {{z1, z2, z3}, {w, i, t}}. This K3,3 is the contraction of a K3,3 of the original input graph H .
FIGURE 1.16: If w is a C node but at least one of z1, z2, and z3 is a non-neighbor of w, the case can be reduced to that of Figure 1.15 by selecting a neighbor w′ to serve in the role of w.
FIGURE 1.18: If a C node x has four neighbors (a, b, c, d) in that order, such that there are disjoint paths from a and c to i and b and d to an ancestor of i, then the cycle that x represents, together with these paths, gives a K3,3.
FIGURE 1.2: An interval graph is the intersection graph of a set of intervals on a line. There is one vertex for each of the intervals, and two vertices are adjacent if and only if the corresponding intervals intersect.
FIGURE 1.6: Assigning a new zero column x to a matrix, computing the PC tree for it, and then picking the PC tree up at x to root it, gives the PQ tree for the matrix, rooted at y, when the C nodes are reinterpreted as Q nodes.
FIGURE 1.9: An order-preserving contraction of an edge xy. The neighbors of x and y are cyclically ordered. The edge is removed and x and y are identified, so that the cyclic order of neighbors of x and y about the edge is preserved.
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