Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Connected-homomorphism-homogeneous graphs   A relational structure is (connected-)homogeneous if every isomorphism
between finite (connected) substructures extends to an automorphism of the
structure. We investigate notions which generalise (connected-)homogeneity,
where "isomorphism" may be replaced by "homomorphism" or "monomorphism" in the
definition. Specifically, we study the classes of finite
connected-homomorphism-homogeneous graphs, with the aim of producing
classifications. The main result is a classification of the finite C-HH graphs,
where a graph G is C-HH if every homomorphism from a finite connected induced
subgraph of G into G extends to an endomorphism of G. The finite C-II
(connected-homogeneous) graphs were classified by Gardiner in 1976, and from
this we obtain classifications of the finite C-HI and C-MI finite graphs.
Although not all the classes of finite connected-homomorphism-homogeneous
graphs are completely characterised, we may still obtain the final hierarchy
picture for these classes.

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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph