# An Introduction to Clique Minimal Separator Decomposition

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- First, not every graph has a clique separator.
- Second, the subgraphs obtained are not disjoint, which means that if the overlap is large, little is to be gained by a Divide-and-Conquer approach.

- Section 2 gives some general graph notions, some particulars on minimal separation and minimal triangulation.
- Section 3 provides a precise definition of clique minimal separator decomposition, as well as examples; we also discuss how to find the clique minimal separators of a graph efficiently, and give some properties of this decomposition.
- Section 4 gives an efficient process for computing the clique minimal separator decomposition of a graph.
- Section 5 provides a brief history of clique decomposition, with the bibliographic background as well as high-level proofs of the results we present in this paper.
- We conclude in Section 6.

## 2. Graph Notions

#### 2.1. General notions

**Neighborhoods:**

**Paths**

**and cycles:**

**Connexity:**

**Cliques:**

**Perfection:**

**Example**

**2.1**

#### 2.2. Minimal separation, chordal graphs, and minimal triangulation

**Definition**

**2.2**

- A subset S of vertices of a connected graph G is called a separator (or sometimes a cutset) if $G(V-S)$ is not connected.
- A separator S is called an $ab$-separator if a and b lie in different connected components of$G(V-S)$, a minimal $ab$-separator if S is an $ab$-separator and no proper subset of S is an $ab$-separator.
- A separator S is a minimal separator, if there is some pair $\{a,b\}$ such that S is a minimal $ab$-separator.

**Example**

**2.3**

**Example**

**2.4**

**Figure 3.**This graph has an exponential number of minimal separators, which are the combinations from the Cartesian product $\{{a}_{1},{b}_{1}\}\times \{{a}_{2},{b}_{2}\}\times \dots \times \{{a}_{n},{b}_{n}\}$.

**Chordal**

**graphs:**

**Figure 4.**Chordal graph H, with minimal separators: $\{b,k\},\{c,k\},\{d,j,k\},\{j,k\},\{i,j\},$ $\{d,g\}.$

**Perfect**

**elimination orderings:**

**Definition**

**2.5**

**Minimal triangulations:**

**Definition**

**2.6**

- minimal if for no proper subset ${F}^{\prime}$ of F, ${H}^{\prime}=(V,E+{F}^{\prime})$ is chordal.
- minimum if no other minimal triangulation has less fill edges.

**Minimal**

**elimination orderings:**

## 3. Defining clique minimal separator decomposition

#### 3.1. Definitions and examples

**Decomposition**

**Step 3.1**

**Property**

**3.2**

**Characterization**

**3.3**

**Example**

**3.4**

- Figure 6 gives the decomposition of a tree.
- On any chordal graph, this decomposition yields the set of maximal cliques of the input graph.

**Figure 6.**(a) A decomposition step using minimal separator $\left\{a\right\}$. The minimal separators (except for $\left\{a\right\}$) are partitioned into the two subtrees obtained. (b) Total decomposition.

#### 3.2. Properties of the atoms

- The number of atoms is at most n.
- The intersection between atoms is always a clique (which may be empty), and it is not necessarily a minimal separator (even when it is not empty).

**Example**

**3.5**

#### 3.3. An equivalent process

**Property**

**3.6**

**Decomposition**

**Step 3.7**

#### 3.4. How to compute the clique minimal separators

**Property**

**3.9**

- compute a minimal triangulation H of G.
- compute the minimal separators of H.
- check each minimal separator of H to see whether it is a clique in G.

#### 3.5. Some problems which can be solved using the atoms

- Minimal and minimum triangulation: if a minimal triangulation is computed for each of the atoms of the clique minimal separator decomposition of a graph, then the union of the set of fill edges obtained defines a minimal triangulation for G. Thus if for each atom a minimum-sized fill is computed, the resulting fill will be a minimum triangulation of G, since the sets of fill edges in the atoms are pairwise disjoint.
- Treewidth: the treewidth is obtained by taking the largest treewidth over all the atoms.
- Perfection: Any chordless cycle of length 4 or more is preserved by a decomposition step, as well as any antihole, so this decomposition preserves holes and antiholes. Thus a graph is perfect if and only if all its atoms are perfect.
- Coloring: An optimal coloring is obtained by merging optimal colorings of the atoms.
- Maximum clique: Any maximal clique of the graph is preserved after a decomposition step, so finding the size of the largest clique of each atom will yield the size of the largest clique of the original graph.

## 4. Algorithms and implementations

## 5. Theoretical and bibliographical background

#### 5.1. A brief history of clique minimal separator decomposition

#### 5.2. Computing the clique minimal separators of a graph

**Property**

**5.1**

**Proof:**

**Corollary**

**5.2**

#### 5.3. Decomposing a graph into atoms

**Theorem**

**5.3**

**Proof:**

#### 5.4. The unicity of clique minimal separator decomposition

**Decomposition**

**Step 5.4**

**Figure 9.**A graph with different decompositions by clique separators. (a) Graph. (b) One decomposition, which is the unique decomposition by clique minimal separators. (c) Another decomposition.

**Definition**

**5.5**

**Property**

**5.6**

**Proof:**

**Characterization 5.7**

**Proof:**

**Corollary**

**5.8**

## 6. Conclusions

## References

- Didi Biha, M.; Kaba, B.; Meurs, M.J.; SanJuan, E. Graph decomposition approaches for terminology graphs. Proc. MICAI
**2007**, 883–893. [Google Scholar] - Kaba, B.; Pinet, N.; Lelandais, G.; Sigayret, A.; Berry, A. Clustering gene expression data using graph separators. In Silico Biol.
**2007**, 7, 433–452. [Google Scholar] [PubMed] - Dirac, G.A. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg
**1961**, 25, 71–76. [Google Scholar] [CrossRef] - Rose, D.J. Triangulated graphs and the elimination process. J. Math. Anal. Appl.
**1970**, 32, 597–609. [Google Scholar] [CrossRef] - Berry, A.; Pogorelcnik, R. A simple algorithm to generate the minimal separators of a chordal graph; Research Report LIMOS RR-10-04; LIMOS UMR CNRS: Aubière, France, 2010. [Google Scholar]
- Yannakakis, M. Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discrete Method
**1981**, 2, 77–79. [Google Scholar] [CrossRef] - Kratsch, D.; Spinrad, J. Minimal fill in O(n
^{2.69}) time. Discrete Math.**2006**, 306, 366–371. [Google Scholar] [CrossRef] - Heggernes, P.; Telle, J.A.; Villanger, Y. Computing minimal triangulations in time O(n
^{α}logn)=o(n^{2.376}). SIAM J. Discrete Math.**2005**, 19, 900–913. [Google Scholar] [CrossRef] - Berry, A.; Blair, J.R.S.; Heggernes, P. Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica
**2004**, 39, 287–298. [Google Scholar] [CrossRef] - Berry, A.; Bordat, J.P.; Heggernes, P.; Simonet, G.; Villanger, Y. A wide-range algorithm for minimal triangulation from an arbitrary ordering. J. Algor.
**2006**, 58, 33–66. [Google Scholar] [CrossRef] - Berry, A.; Heggernes, P.; Villanger, Y. A vertex incremental approach for maintaining chordality. Discrete Math.
**2006**, 306, 318–336. [Google Scholar] [CrossRef] - Heggernes, P. Minimal triangulations of graphs: A survey. Discrete Math.
**2006**, 306, 297–317. [Google Scholar] [CrossRef] - Rose, D.J.; Tarjan, R.E.; Lueker, G.S. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput.
**1976**, 5, 266–283. [Google Scholar] [CrossRef] - Berry, A.; Krueger, R.; Simonet, G. Maximal label search algorithms to compute perfect and minimal elimination orderings. SIAM J. Discrete Math.
**2009**, 23, 428–446. [Google Scholar] [CrossRef] - Tarjan, R.E. Decomposition by clique separators. Discrete Math.
**1985**, 55, 221–232. [Google Scholar] [CrossRef] - Gavril, F. Algorithms on clique separable graphs. Discrete Math.
**1977**, 19, 159–165. [Google Scholar] [CrossRef] - Whitesides, S. An algorithm for finding clique cutsets. Inf. Process. Lett.
**1981**, 12, 31–32. [Google Scholar] [CrossRef] - Dahlhaus, E.; Karpinski, M.; Novick, M.B. Fast parallel algorithms for the clique separator decomposition. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms: SODA ’90, Philadelphia, PA, USA, January 1990; pp. 244–251.
- Leimer, H.G. Optimal decomposition by clique separators. Discrete Math.
**1993**, 113, 99–123. [Google Scholar] [CrossRef] - Olesen, K.G.; Madsen, A.L. Maximal prime subgraph decomposition of Bayesian networks. IEEE Trans. Syst. Man Cybernet. B
**2002**, 32, 21–31. [Google Scholar] [CrossRef] [PubMed] - Blair, J.R.S.; Peyton, B.W. An introduction to chordal graphs and clique trees. Graph Theory Sparse Matrix Comput.
**1993**, 84, 1–29. [Google Scholar] - Brandstädt, A.; Hoàng, C.T. On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem. Theor. Comput. Sci.
**2007**, 389, 295–306. [Google Scholar] [CrossRef] - Bodlaender, H.L.; Koster, A.M.C.A. Safe separators for treewidth. Discret Math.
**2006**, 306, 337–350. [Google Scholar] [CrossRef]

© 2010 by the authors. Licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Berry, A.; Pogorelcnik, R.; Simonet, G.
An Introduction to Clique Minimal Separator Decomposition. *Algorithms* **2010**, *3*, 197-215.
https://doi.org/10.3390/a3020197

**AMA Style**

Berry A, Pogorelcnik R, Simonet G.
An Introduction to Clique Minimal Separator Decomposition. *Algorithms*. 2010; 3(2):197-215.
https://doi.org/10.3390/a3020197

**Chicago/Turabian Style**

Berry, Anne, Romain Pogorelcnik, and Geneviève Simonet.
2010. "An Introduction to Clique Minimal Separator Decomposition" *Algorithms* 3, no. 2: 197-215.
https://doi.org/10.3390/a3020197