The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma ca...

The global forcing number of a graph G is the minimal cardinality an edge subset discriminating all perfect matchings G, denoted by gf(G). For matching M S⊆E(G)∖M such that G−S has unique called anti-forcing M. maximum among Af(G). It known hexagonal system equals famous Fries number. bipartite we show gf(G)≥Af(G). Next extend result to Birkhoff–von Neumann graphs, whose polytopes are character...

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

The class forcing theorem, which asserts that every class forcing notion P admits a forcing relation P, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel-Bernays set theory GBC to the principle of elementary transfinite recursion ETROrd for class...

A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u–v path of G, where u, v 2 D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing ...

For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdx(...

For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S i...

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...