For a family scriptF of graphs, a graph G is scriptF‐saturated if G contains no member of scriptF as a subgraph, but for any edge uv in G¯, G+uv contains some member of scriptF as a subgraph. The minimum number of edges in an scriptF‐saturated graph of order n is denoted sat(n,F). A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let S(H) denote the family of H‐subdivisions, including H itself. In this paper, we study sat(n,S(H)) when H is one of Ct or Kt, obtaining several exact results and bounds. In particular, we determine sat(n,S(Ct)) exactly for 3≤t≤5 and show for n sufficiently large that there exists a constant ct such that 54n≤sat(n,scriptS(Ct))≤54+cttn. For t≥36 we show that ct=8 will suffice, and that this can be improved slightly depending on the value of t(mod8). We also give an upper bound on sat(n,S(Kt)) for all t and show that sat(n,scriptS(K5))=⌈3n+42⌉. This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge‐maximal graphs without a K5‐minor have at least 11n6 edges.
We prove that if a directed multigraph D has at most t pairwise arc-disjoint directed triangles, then there exists a set of less than 2t arcs in D which meets all directed triangles in D, except in the trivial case t = 0. This answers affirmatively a question of Tuza from 1990.
We define a new, partizan, loopy combinatorial game, Forced-Capture Hnefatafl, similar to Hnefatafl, except that players are forced to make capturing moves when available. We show that this game is PSPACE-hard using a reduction from Constraint Logic, making progress towards classifying proper Hnefatafl.
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