Summary: Two problems on the class of \(k\)-trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An \(O(\log^2 n)\) time parallel algorithm for the fast reordering problem is described that uses \(O(nk(n- k)/\log n)\) processors on a CRCW PRAM proving membership in the class \({\mathcal N}{\mathcal C}\) for fixed \(k\). An \(O(nk(k+ 1)!)\) time sesquential algorithm for the isomorphism problem is obtained representing an improvement over the \(O(n^2 k(k+1)!)\) algorithm of Sekharan (the second author). A parallel version of this sequential algorithm is presented that runs in \(O(\log^2 n)\) time using \(O((nk((k+ 1)!+ n-k))/\log n)\) processors improving on a parallel algorithm of Sekharan for the isomorphism problem. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a \(k\)-tree.