The goal of this paper is to extend to infinite graphs the known Morse inequalitiesfor discrete Morse functions proved by R. Forman in the finite case. In order to getthis result we shall use a special kind of infinite subgraphs on which a discrete Morsefunction is monotonous, namely, decreasing rays. In addition, we shall use this resultto characterize infinite graphs by the number of critical elements of discrete Morsefunctions defined on them.
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