The Clique


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We consider the following generalization of recognition problems. Graph sandwich problems were defined in the context of Computational Biology and have attracted much attention lately arising from many applications and as a natural generalization of recognition problems [4, 7, 8, 9, 11]. Golumbic et al. Clique graphs admit a characterization by a special edge cover. A complete set C covers the edge uv if the endvertices u and v belong to C. A complete edge cover of G is a family of complete sets covering all edges of G. Theorem 1 Roberts and Spencer [16] A graph G is a clique graph if and only if there exists a complete edge cover of Gsatisfying the Helly property.

If G admits an RS -family, then G admits a small RS -family with at most E sets, because every edge needs to be covered by only one complete set and any subfamily of a family satisfying the Helly property does satisfy the Helly property too. Therefore, the characterization given in Theorem 1 gives a short certificate for the class of clique graphs because the Helly property for a family of subsets can be tested in polynomial time on the size of the family [2, 16, 18]. On the other hand, recognizing clique graphs was recently shown to be NP -complete [1]. The family of all cliques of G is a complete edge cover of G , so it follows that clique-Helly graphs are clique graphs.


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A polynomial-time recognition algorithm for the class of clique-Helly graphs was presented in [6] and independently in [17]. This algorithm is a consequence of a characterization of clique-Helly graphs based on the concept of extended triangle.

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A triangle T of a graph G is a complete set containing exactly three vertices. The extended triangle of G relative to the triangle T is defined in [17] as the subgraph induced in G by the vertices adjacent to at least two vertices of T and it is denoted T '. A universal vertex in a graph is adjacent to every other vertex of the graph.

Theorem 2 Dragan [6], Szwarcfiter [17] A graph G is a clique-Helly graph if and only if every extended triangle of G contains a universal vertex. A polynomial-time recognition algorithm for the class of hereditary clique-Helly graphs was presented in [13]. This algorithm uses a finite family of induced forbidden subgraphs, the so-called ocular graphs depicted in Figure 1.

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Ocular graphs were defined in [19] as a tool to study the Helly property and were shown in [13] to be the minimal forbidden configurations for hereditary clique-Helly graphs. Theorem 3 Prisner [13] A graph G is a hereditary clique-Helly graph if and only if G contains none of the four graphs shown in Figure 1 as an induced subgraph.

Clearly, from Theorems 2 and 3 one can also recognize in polynomial time whether a graph is clique-Helly nonhereditary.

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Surprisingly, we obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the corresponding sandwich problem is NP -complete. Additionally, we propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.

The studies on sandwich problems focus on properties that are neither hereditary nor ancestral on subgraphs, in which case the complexity of the recognition problem and of the corresponding sandwich problem coincide. The graph classes considered in the present paper: clique, clique-Helly, hereditary clique-Helly are neither hereditary nor ancestral on subgraphs.

Most properties considered so far are hereditary on induced subgraphs, which means that the corresponding graph class admits a family of minimal forbidden subgraphs. In particular, every studied sandwich problem corresponding to a graph class defined by a finite family of minimal forbidden subgraphs has been classified as Polynomial. Note the classes clique and clique-Helly are not hereditary on induced subgraphs. The class hereditary clique-Helly is the first example of a graph class defined by a finite family of minimal forbidden subgraphs for which the corresponding sandwich problem is NP -complete.

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These two decision problems are defined as follows. Question: Is there a truth assignment for X such that each clause in C has at least one true literal? We assume without loss of generality that each variable occurs both as a positive and as a negative literal. For each variable x i , add to V four variable vertices : X i , i , a i and b i. The set V contains 4 n variable vertices, plus 3 m clause vertices.

Please refer to Figure 2 , where we have depicted the forced edges as continuous edges, and the optional edges as dotted edges. Each triple of clause vertices corresponding to a clause cj induce a triangle T j in G 1 , called clause triangle. Vertices a i , b i , X i , i induce a forced variable gadget D i in G 1 , composed by two triangles that share one edge X i i. Let y jr , y js , y jt be the clause vertices of a clause triangle T j.


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The construction of the particular instance V, E 1 , E 2 is concluded. Furthermore, we show there are only two cases for a triangle to yield an ocular graph. Please refer to Figure 2. We shall use the characterization of Theorem 3.

senjouin-renshu.com/wp-content/53/2487-mapa-radares-moviles.php Suppose first that C is satisfiable, and consider a truth assignment for X such that each clause c j in C has at least one true literal. To define the sandwich graph G , if x i has value true, add to E , for every clause c j where xi occurs as positive literal, the optional edge y ji X i ; if x i has value false, add to E , for every clause c j where x i occurs as negative literal, the optional edge y ji i. Note that in the constructed sandwich graph G , for each variable x i , either all optional edges incident to vertex X i or all optional edges incident to vertex i are present.

Conversely, suppose that G is a hereditary clique-Helly sandwich graph for the particular constructed instance V, E 1 , E 2. It follows that G contains none of the forbidden configurations depicted in Figure 1. Therefore we can construct a truth assignment by setting variable x i as true if there exists in G an edge y ji X i for some clause c j where x i occurs as positive literal, and by setting variable x i as false otherwise.

Figure 3 shows an example of an instance constructed in Theorem 4. We have depicted the forced edges as continuous edges, and the optional edges as dotted edges. In order to establish the hardness of the problem, we perform the same reduction from 3-SAT of Theorem 4. It is sufficient to show that every sandwich graph of V, E 1 , E 2 which is clique-Helly is also hereditary clique-Helly. By Theorem 3, G has as induced subgraph one of the graphs of Figure 1. Denote it by G'. In both cases there are two variable vertices v 1 and v 2 in G'. See our cookie policy Accept cookies.

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