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Motivated by the problem of enumerating all tree decompositions of a graph,we consider in this article the problem of listing all the minimal chordalcompletions of a graph. In \cite{carmeli2020} (\textsc{Pods 2017}) Carmeli\emph{et al.} proved that all minimal chordal completions or equivalently allproper tree decompositions of a graph can be listed in incremental polynomialtime using exponential space. The total running time of their algorithm isquadratic in the number of solutions and the existence of an algorithm whosecomplexity depends only linearly on the number of solutions remained open. Weclose this question by providing a polynomial delay algorithm to solve thisproblem which, moreover, uses polynomial space.  Our algorithm relies on \emph{Proximity Search}, a framework recentlyintroduced by Conte \emph{et al.} \cite{conte-uno2019} (\textsc{Stoc 2019})which has been shown powerful to obtain polynomial delay algorithms, butgenerally requires exponential space. In order to obtain a polynomial spacealgorithm for our problem, we introduce a new general method called\emph{canonical path reconstruction} to design polynomial delay and polynomialspace algorithms based on proximity search.

Polynomial delay algorithm for minimal chordal completions