Graphical properties of polyhexes: Perfect matching vector and forcing

Forcing, Freedom, & Uniqueness in Graph Theory & Chemistry

“…A further related definition considers subsets 6 e S with members which are either edges or 6-cycles of G, so that 6 e S is partitioned into a set e S of edges of G and a set 6 S of 6-cycles of G. 6 S is conjugated in κ , and no other Kekule structure has this same relation to 6 e S . The e6-forcing number of κ of G is the minimum size of such an 6 e S .…”

“…[37][38][39][40][41][42][43] Next, our initial view of "forcing" is indicated. 6 S of conjugated 6-cycles of G is 6-forcing if no other Kekule structure κ of G manifests the same subset of conjugated 6-cycles with the same conjugation pattern within each of these cycles. For each of the sets τ S  e S , ( ) e S , or 6 S with ,( ), or 6 τ e e  , define the τ -freedom of κ as the minimum order of a τ -forcing of κ .…”

“…6 S of conjugated 6-cycles of G is 6-forcing if no other Kekule structure κ of G manifests the same subset of conjugated 6-cycles with the same conjugation pattern within each of these cycles. For each of the sets τ S  e S , ( ) e S , or 6 S with ,( ), or 6 τ e e  , define the τ -freedom of κ as the minimum order of a τ -forcing of κ . The sum of τ -freedoms for the Kekule structures of G gives a net τ -force ( ) τ f G , which evidently is a graph invariant, with…”

“…[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] But it should be mentioned that not all Kekule structures of general graphs admit a 6-forcing set, and consequently do not admit a 6-freedom. Such clearly is the circumstance with any multi-Kekulean graph without any 6-cycles -e.g., the 4-cycle (butadiene).…”

“…He encouraged collaboration on the topic, which indeed ensued: a degree of concurrence of opinions was achieved, some initial results were established (on forcing, and some other ideas), and a paper subsequently appeared. 6 Following this work, forcing for Kekule structures has continued to receive attention, in both the mathematical, [7][8][9][10][11][12][13][14][15][16][17][18] and the chemical [19][20][21][22][23][24][25][26][27] literature. And further, what has been termed an "anti-forcing" idea has also been proposed 17 & studied.…”