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The complexity of clique graph recognition

Profile image of Marisa GUTIERREZMarisa GUTIERREZ

2009, Theoretical Computer Science

https://doi.org/10.1016/J.TCS.2009.01.018
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12 pages

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Abstract

Consider finite, simple and undirected graphs. V and E denote the vertex set and the edge set of the graph G, respectively. A complete set of G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. The clique family of G is denoted by C(G). The clique graph of G is the intersection graph of C(G). The clique operator, K, assigns to each graph G its clique graph which is denoted by K(G). On the other hand, say that G is a clique graph if G belongs to the image of the clique operator, i.e. if there exists a graph H such that G = K (H). Clique operator and its image were widely studied. First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I<n, for some positive integer n; and the behavior under these operators of parameters such as the number of vertices or diameter were studied in [5,8,9,12,26,30] and more recently in [7,14,21-23, 29], For several classes of graphs, the image of the class under the clique operator was characterized [10,18,19,24,34,37]; and, in some cases, also the inverse image of the class [16,28,35], Results of the previous bibliography can be found in the survey [39], Clique graphs have been much studied as intersection graphs and are included in several books [11,25,33], In this paper we are concerned with the time complexity of the problem of recognizing clique graphs, this is the time complexity of the following decision problem.

FAQs

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AI

What key theorem establishes the NP-completeness of clique graph recognition?add

The main theorem proves that clique graphs are NP-complete via a reduction from the 3-satisfiability problem.

How does the constructed graph G( relate to the satisfiability of clauses?add

The graph G( is a clique graph if and only if the set of clauses is satisfiable.

What is the significance of RS-families in the recognition of clique graphs?add

An RS-family allows for a polynomial verification of whether a graph is a clique graph.

What distinguishes clique graphs from clique-Helly graphs in this study?add

Clique graphs do not necessarily have the Helly property, unlike clique-Helly graphs, indicating a complexity distinction.

How do bounded parameters influence the complexity of clique graph recognition?add

The NP-completeness holds for graphs with bounded clique size greater than 12 and maximum degree 14.

Figures (8)
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
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
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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