A compact graph-like space is a triple (X, V, E) where X is a compact, metrizable space, V ⊆ X is a closed zero-dimensional subset, and E is an index set such that X \ V ∼ = E × (0, 1). New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.2010 Mathematics Subject Classification. 05C38, 05C45, 05C63, 54F15, 57M15.

A space is n-arc connected (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ 0 -ac are characterized. The complexity of characterizing n-ac graphs for n = 2, 3, 4, 5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.2010 Mathematics Subject Classification. Primary 54F15.; Secondary 54D05, 54F15, 54H05.

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A surjective continuous map f : [0, 1] → X is called an arcwise increasing map if for any two closed subintervalsA continuum X is said to admit an arcwise increasing map if there is an arcwise increasing map onto X. It is shown that any Peano continuum with no free arcs admits an arcwise increasing map, and a characterization of graphs and dendrites that admit arcwise increasing maps is given.

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