On the complexity of tree embedding problems

Embedding of cycles and wheels into arbitrary trees

“…From the above NP-complete results, the dilation-sum problem is expected to be harder than the dilation problem [27]. That is why, even though there are numerous results and discussions on the dilationsum problem and the congestion-sum problem, most of them deal with only approximation results.…”

“…The dilation problem is NP-complete for two classes of 'almost' catepillars on a path [18] and trees of maximum degree 3 on paths [20,27]. From the above NP-complete results, the dilation-sum problem is expected to be harder than the dilation problem [27].…”

Embedding of C_n^2 and C_(n-1)^2+K_1 in to Arbitrary Tree

“…The dilation-sum problem of an arbitrary graph on a path is called the linear layout or the linear arrangement problem in the VLSI literature [20]. The concept of embedding is widely studied in the literature of fixed interconnection parallel architectures [21].The dilation problem is NP-complete for two classes of'almost' catepillars on a path [18] and trees of maximum degree 3 on paths [20,27]. From the above NP-complete results, the dilation-sum problem is expected to be harder than the dilation problem [27].…”

“…The concept of embedding is widely studied in the literature of fixed interconnection parallel architectures [21].The dilation problem is NP-complete for two classes of'almost' catepillars on a path [18] and trees of maximum degree 3 on paths [20,27]. From the above NP-complete results, the dilation-sum problem is expected to be harder than the dilation problem [27]. That is why, even though there are numerous results and discussions on the dilationsum problem and the congestion-sum problem, most of them deal with only approximation results.…”

“…The bounded cost of dilation and congestion has been estimated for the embedding on binary trees [27]. Most of the work on the dilation-sum problem and the dilation problem are for the particular case in which the host graph is a path, or a cycle [20].The concept of cutwidth is a special case of congestion when the host graph is a path [11,26,29].…”