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More precisely, is there an undirected graph X with a vertex v such that if we construct a graph G 2(X) by taking two copies of X and connecting their v vertices by an edge, and a graph G 1(X) by adding two new vertices to X andG w ttype td c m x morris cowley 1928 x j e c m x p plymouth 1949 v x a j r renault 35cv 1911 g ^ @ x v gt iesso toyota team lemans j {w whippet 1926 e b y b gG Therefore, they must be adjacent in G Hence there exists a u;vpath in G If u and v belong to the same connected component of G and w is a vertex in a di erent connected component of G, then both u and v are adjacent w in G If e 1 is the edge of G with endvertices u and w and e 2 is the edge of G with endvertices w and v, then (u;e 1;w;e
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A multigraph is a pair G= (V;E) where V is a nite set and Eis a multiset of multigraph elements from V 1 V 2, ie, we also allow loops and multiedges A hypergraph is a pair H= (X;E) where Xis a nite set and E 2f;g hypergraph De nition For two graphs G 1 = (V 1;E 1) and G 2 = (V 2;E 2) we say that G 1 and G 2 are isomorphic, denoted by GU Ђ̒ ̎d ߂ ̈ ʌ ́A w 菇 x ƂȂ ܂ B ʂ̉ Ђʼn Ђ̎d ߂ w 菇 x Ƃ ̂ɂ Ă Ђ͏ Ȃ 悤 ł B ǂ̉ Ђł ꂼ Ǝ ̌ŗL Ă ܂ B w K x ƌĂԂƂ 낪 ł ˁB Ȃ ǂݕ ͓ w Ă x ł w K x Ƃ g ЂƁA w K x Ƃ g Ђ ܂ B Ђɂ Ă͏ ʂ̋K w K x Ƃ āA ʂ̋K w K x Ƃ āA ꂼ w قǁx A w x Ɠǂݕ Ă Ђ ܂ B Ȃ w K x ͓ Łw K x ͖ Ȃ̂ŁA w K x Ƃ 邪 Ƃ l ܂ BIf f and g are the functions whose graphs are shown, let u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = g(g(x)) Find each derivative, if it exists If it does not exist, explain why
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