.. .. Let-l(t-k?1-;-=-{j-?-{2, has no child clique yet} be the set of leaves of T k?1. Then, C k is connected to a leaf C j * ? L(T k?1 ) and C j * ? C k = ?, see the proof of Lemma 21 below

, if ?i ? B(T k?1 ) such that C i ?C k = ?, ? j ? {i + 1,. .. , k ? 1} such that (C i , C j ) ? E k?1 and C j ? C k = ?, then ? j * ? L(T k?1 ) such that C j * ? C k = ?. In words, if any branch clique of T k?1 cannot be chosed as a parent clique for C k without violating the DIP

.. .. {1, ?. 1}-|-c-i-?-c-k, and =. ?}, By connectivity, I(C k ) = ?. Thus, if I(C k ) ? B(T k?1 ) = ?, then our first remark yields that I(C k ) ? L(T k?1 ) = ? : now consider the case in which I(C k ) ? B(T k?1 ) = ?

, latest clique satisfying C i * ? C k = ? which is not a leaf of T k?1. Since i * ? I(C k ) ? B(T k?1 ), by assumption there is a j * ? {i * + 1,. .. , k ? 1} such that (C i * , C j * ) ? E k?1 and C j * ? C k = ?. By maximality of i *

B. Bollobás, Volume estimates and rapid mixing. Pages 151-180 in Flavors of geometry, vol.31, 1997.

M. E. Dyer and A. M. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Comput, vol.17, pp.967-974, 1988.

G. Elekes, A geometric inequality and the complexity of measuring the volume, Discrete Comput. Geom, vol.1, pp.289-292, 1986.

D. Henrion, J. B. Lasserre, and C. Savorgnan, Approximate volume and integration for basic semialgebraic sets, SIAM Review, vol.51, issue.4, pp.722-743, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00297384

B. Büeler, A. Enge, and K. Fukuda, Exact volume computation for polytopes: a practical study, Pages 131-154 in Polytopes: combinatorics and computation, 2000.

M. E. Dyer, A. M. Frieze, and R. Kannan, A random polynomial-time algorithm for approximating the volume of convex bodies, J. ACM, vol.38, pp.1-17, 1991.

C. Belisle, Slow hit-and-run sampling, Statist. Probab. Lett, vol.47, pp.33-43, 2000.

C. Belisle, E. Romeijn, and R. L. Smith, Hit-and-run algorithms for generating multivariate distributions, Math. Oper. Res, vol.18, pp.255-266, 1993.

R. L. Smith, Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions, Oper. Res, vol.32, pp.1296-1308, 1984.

B. Cousins and S. Vempala, A practical volume algorithm, Math. Program. Comput, vol.8, pp.133-160, 2016.

J. B. Lasserre, Moments, positive polynomials and their applications, 2010.

J. B. Lasserre, Computing Gaussian and exponential measures of semi-algebraic sets, Adv. Appl. Math, vol.91, pp.137-163, 2017.

D. Henrion and M. Korda, Convex computation of the region of attraction of polynomial control systems, IEEE Trans. Autom. Control, vol.59, issue.2, pp.297-312, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00723019

C. Josz, D. K. Molzahn, M. Tacchi, and S. Sojoudi, Transient stability analysis of power systems via occupation measures, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02125599

J. B. Lasserre, Convergent SDP relaxations in polynomial optimization with sparsity, SIAM J. Optim, vol.17, pp.822-843, 2006.

D. Henrion, J. B. Lasserre, and J. Loefberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optimization Methods and Software, vol.24, issue.4-5, pp.761-779, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00172442

J. R. Blair and B. Peyton, An introduction to chordal graphs and clique trees. Pages 1-29 in Graph Theory and Sparse Matrix Computation, 1993.

J. B. Lasserre, Representation of chance-constraints with strong asymptotic guarantees, IEEE Control Systems Letters, vol.1, issue.1, pp.50-55, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01487006

J. H. Hubbard and B. B. Hubbard, Vector calculus, linear algebra, and differential forms-A unified approach, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01297648

R. Stanley, I. G. Macdonald, and R. B. Nelsen, Solution of elementary problem E2701, The American Mathematical Monthly, vol.86, issue.5, p.396, 1979.