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Fig. 1. (a) Non-clique graph thus non-clique-Helly graph; (b) non-clique-Helly graph but clique graph; (c) clique-Helly graph thus clique graph. CLIQUE GRAPH

Figure 1 (a) Non-clique graph thus non-clique-Helly graph; (b) non-clique-Helly graph but clique graph; (c) clique-Helly graph thus clique graph. CLIQUE GRAPH

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Fig. 3. Satisfaction Testing component 5; for clause c; with: (a) 2 literals corresponding to variables t,, and u,, ; and (b) 3 literals corresponding to variable: uj,, Uj, and u,,. Vertices {@, dj li, gyn a i, | i € Ij} induce a complete graph on 12 vertices but for simplicity some edges are not drawn.
Fig. 3. Satisfaction Testing component 5; for clause c; with: (a) 2 literals corresponding to variables t,, and u,, ; and (b) 3 literals corresponding to variable: uj,, Uj, and u,,. Vertices {@, dj li, gyn a i, | i € Ij} induce a complete graph on 12 vertices but for simplicity some edges are not drawn.
Fig. 4. Graph G, obtained from the 3sary instance J = (U,C), U = {niy, tig. ta}, C = ({ty. my), (tt. ug, 1a}. (ty, ta}.
Fig. 4. Graph G, obtained from the 3sary instance J = (U,C), U = {niy, tig. ta}, C = ({ty. my), (tt. ug, 1a}. (ty, ta}.
Fig. 5. Figure used inthe proof of Lemma 6. Any RS-family of G; must contain the highlighted triangles, In order to cover highlighted edge GC: ,exactly one of the triangles {a , c., a; } or {a;,, c} , d,} must belong to F.
Fig. 5. Figure used inthe proof of Lemma 6. Any RS-family of G; must contain the highlighted triangles, In order to cover highlighted edge GC: ,exactly one of the triangles {a , c., a; } or {a;,, c} , d,} must belong to F.
Fig. 6. Figure used in the proof of Lemma 7. For any variable uj andj € J;, any RS-family of G; must contain either the filled triangles in (a) or the fillec triangles in (b). In any case the bold triangles must belong to the RS-family.
Fig. 6. Figure used in the proof of Lemma 7. For any variable uj andj € J;, any RS-family of G; must contain either the filled triangles in (a) or the fillec triangles in (b). In any case the bold triangles must belong to the RS-family.
Fig. 7. Triangles of Ky) (j) which may be not easy, they contain edges covered not only by the Kj) (j) itself but also by other members of F.
Fig. 7. Triangles of Ky) (j) which may be not easy, they contain edges covered not only by the Kj) (j) itself but also by other members of F.
Fig. 8. Triangles of Ks (j, i) which may be not easy, they contain edges covered not only by the Ks (j, f) itself but also by other members of ¥.
Fig. 8. Triangles of Ks (j, i) which may be not easy, they contain edges covered not only by the Ks (j, f) itself but also by other members of ¥.
Fig. 9. RS-cover ¥ for graph G, of Fig. 4. The RS-cover is defined by the satisfying truth assignment where uj, is true, and u) and uz are false. Bold edges highlight forced triangles present in every RS-cover of G;. Filled connected regions depict triangles of # which depend on the truth assignment for! = (U, C), Complete sets Ks (j, i) and Kj»{/) belong to the RS-cover ¥ but are not depicted in order to make simpler the drawing.
Fig. 9. RS-cover ¥ for graph G, of Fig. 4. The RS-cover is defined by the satisfying truth assignment where uj, is true, and u) and uz are false. Bold edges highlight forced triangles present in every RS-cover of G;. Filled connected regions depict triangles of # which depend on the truth assignment for! = (U, C), Complete sets Ks (j, i) and Kj»{/) belong to the RS-cover ¥ but are not depicted in order to make simpler the drawing.
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MathematicsComputer ScienceCombinatoricsTheoretical Computer ScienceNP-Complete Problem
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