The unpath number of a graph 
(Akiyama et al. 1980, p. 415), called the apathy
of 
by Harary and Palmer (1973, p. 268),
is the maximum number of edge-disjoint connected
subgraphs of 
that are not paths. It is the packing counterpart of
the path number, in the same covering/packing sense
that anarboricity is the packing counterpart of
arboricity.
Unpath Number
See also
Anarboricity, Path NumberExplore with Wolfram|Alpha
References
Akiyama, J.; Exoo, G.; and Harary, F. "Covering and Packing in Graphs III: Cyclic and Acyclic Invariants." Math. Slovaca 30, 405-417, 1980.Harary, F. and Palmer, E. M. "A Survey of Graphical Enumeration Problems." In A Survey of Combinatorial Theory (Ed. J. N. Srivastava). Amsterdam, Netherlands: North-Holland, pp. 259-275, 1973.Zelinka, B. "Unpath Number of a Complete Multipartite Graph." Math. Slovaca 33, 293-296, 1983.Cite this as:
Weisstein, Eric W. "Unpath Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnpathNumber.html