. If-|v-(h)|-=-6, , vol.7

, If |V (H)| = 7, then Lemma 5.9 proves that H contains one of A 9, vol.10

. If-|v-(h)|-=-8, , vol.13

, ? 2 ) if for each x 1 y 1 ? E(G) and each x 2 y 2 ? E(H) there exists a homomorphism f : (G 1 , ? 1 ) ? (G 2 , ? 2 ) such that f (x) = w and f (y) = z. A signed graph (G 1 , ? 1 ) is triangle-transitively homomorphic to (G 2 , ? 2 ) if for each unbalanced (respectively, balanced) 3-cycle x 1 y 1 z 1 x 1 in (G 1 , ? 1 ) and each unbalanced (respectively, balanced) 3-cycle

, Recall that ¯ L s is the set of all planar graphs having one of the elements of L s as a spanning subgraph. Let L tri denote the set of all triangulations of the elements of L s. Observe that every element of ¯ L s is a subgraph of an element of L tri

?. ,

, * ) is a signed outerplanar graph. We know that every signed outerplanar graph admits a vertex-mapping to, vol.9

, Thus we can extend the mapping by mapping the vertex v to the vertex ? of (P + 9

, Let v be a vertex of a signed graph (A, ?) having degree at most three such that the neighbors of v induce a clique in A. Then we can extend any homomorphism of (A \ {v}, ?) ? (P + 9

, Finally we handle the signed planar graphs whose underlying graphs are triangulations of A 12 , A 14 and A 15

, Any signed planar graph (A, ?) whose underlying graph is a triangulation of A 12 , A 14 or A 15 admits a homomorphism to

, The graph A 14 ? {bf } is 4-connected, thus we are done by Observation 4.2. On the other hand, the graph A 14 ? {ae} is not 4-connected. Observe that the graph A 14 ? {ae} has exactly two vertices of b and f having degree three each. The neighbors of b and f are cliques of order three. Also if we delete the vertices b and f from A 14 ? {ae}, then the graph we obtain is 4-connected. Therefore, we are done by Observations 4.2 and 4.7. There are exactly four triangulations of A 15 , obtained by adding two edges from {ag, bh} × {ce, df }. The graph A 14 ? {bh, df } is 4-connected, thus we are done by Observation 4.2. On the other hand, the other triangulations have at most four vertices having degree three. The neighbors of each of these vertices are cliques of order three, total, 47 non-isomorphic planar underlying push cliques (1 on 1 vertex, 1 on 2 vertices, 1 on 3 vertices, 3 on 4 vertices, 4 on 5 vertices, 10 on 6 vertices, 14 on 7 vertices and 13 on 8 vertices). See the lists in the webpage

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