Efficient algorithms for combinatorial problems on graphs with bounded, decomposability—a survey
BIT - Ellis Horwood series in artificial intelligence
On search decision and the efficiency of polynomial-time algorithms
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
A separator theorem for graphs with an excluded minor and its applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
On the complexity of finding iso- and other morphisms for partial k-trees
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
The pathwidth and treewidth of cographs
SIAM Journal on Discrete Mathematics
On linear time minor tests with depth-first search
Journal of Algorithms
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
The complexity of subgraph isomorphism for classes of partial k-trees
Theoretical Computer Science
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs
European Journal of Combinatorics
Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
Approximating the Longest Cycle Problem in Sparse Graphs
SIAM Journal on Computing
The Approximation of Maximum Subgraph Problems
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Sequential and Parallel Algorithms for Embedding Problems on Classes of Partial k-Trees
SWAT '94 Proceedings of the 4th Scandinavian Workshop on Algorithm Theory
Which Problems Have Strongly Exponential Complexity?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Finding a Path of Superlogarithmic Length
SIAM Journal on Computing
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Finding paths and cycles of superpolylogarithmic length
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating Maximum Clique by Removing Subgraphs
SIAM Journal on Discrete Mathematics
Finding large cycles in Hamiltonian graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Combinatorica
Note: On the complexity of approximating the Hadwiger number
Theoretical Computer Science
Paired approximation problems and incompatible inapproximabilities
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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We consider the ''minor'' and ''homeomorphic'' analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to K"h or a subgraph homeomorphic to K"h, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(n) bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al. Next, we show that another known result of Bollobas and Thomason and of Komlos and Szemeredi provides an O(n) bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an @W(n^1^/^2^-^O^(^1^/^(^l^o^g^n^)^^^@c^)) lower bound (for some constant @c, unless NP@?ZPTIME(2^(^l^o^g^n^)^^^O^^^(^^^1^^^))) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound. Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.