Canonical forms for cycles in bridge graphs
S. Gill Williamson · Linear and Multilinear Algebra · 1993
Let G = (V,E) be a biconnected graph and let C be a cycle in G. The subgraphs of G identified with the biconnected components of the contraction of C in G are called the bridges of C. Associated with the set of bridges of a cycle C is an auxilliary graphical structure GC called a bridge graph or an overlap graph. Such auxilliary graphs have provided important insights in classical graph theory, algorithmic graph theory, and complexity theory. In this paper, we use techniques from algorithmic combinatorics and complexity theory to derive canonical forms for cycles in bridge graphs. These canonical forms clarify the relationship between cycles in bridge graphs, the structure of the underlying graph G, and lexicographic order relations on the vertices of attachment of bridges of a cycle. The first canonical form deals with the structure of induced bridge graph cycles of length greater than three. Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characterization of minimal elements in certain related posets: ordered bridge three-cycles (10 minimal elements), bridge three-cycles (5 minimal elements), bridge deletion three-cycles (infinite number, 7 classes), minor order (K 5 K 3,3), chordal bridge three-cycles (13 minimal elements), contraction poset (5 minimal elements), cycle-minor poset (infinite number, 14 classes). These results, each giving a different insight into the structure of bridge three-cycles, follow as corollaries from the characterization of the 10 minimal elements of the ordered bridge three-cycle poset. This characterization is constructive and may be regarded as an extension of the classical Kuratowski's Theorem which follows as a corollary. Algorithms are described for constructing these various minimal elements in time O(∣E∣) or O(∣V∣) depending on the case. The first canonical form gives a constructive proof of the result that a graph is nonplanar if and only if it has a cycle C whose bridge graph GC (alternatively, skew bridge graph) has a three-cycle. An algorithm is described that constructs this three-cycle in time O(∣E∣). This is best possible.