On well-quasi-ordering-finite graphs by immersion

Abstract

It has been conjectured by Nash-Williams that the class of all graphs is well-quasi-ordered under the quasi-order ≦ defined by immersion. Two partial results are proved which support this conjecture. (i) The class of finite simple graphsG with\(G \ngeqq K_{2,3} \) K 2,3 is well-quasi-ordered by ≦, (ii) it is shown that a class of finite graphs is well-quasi-ordered by ≦ provided that the blocks of its members satisfy certain restrictive conditions. (In particular, this second result implies that ≦ is a well-quasi-order on the class of graphs for which each block is either complete or a cycle.)

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Andreae, T. On well-quasi-ordering-finite graphs by immersion. Combinatorica 6, 287–298 (1986). https://doi.org/10.1007/BF02579254

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