The first picture below has $v=12$ vertices, $e=16$ vertices, and seems to have $k=4$ crosscaps (denoted by something like $\oplus$). The number of faces $f$ should satisfy $$v-e+f=2-k$$ which gives $f=2$. However looking at the picture we see $f=3$. A possible explanation is that one crosscap is superfluous as the red path shows.

The second picture has no superfluous crosscaps.

My question is, is there a simple way to identify superfluous crosscaps in more complicated situations? My only heuristic so far is that a "loop" of crosscaps has 1 superfluous crosscap.

The first picture below has $v=12$ vertices, $e=16$ edges, and seems to have $k=4$ crosscaps (denoted by something like $\oplus$). The number of faces $f$ should satisfy $$v-e+f=2-k$$ which gives $f=2$. However looking at the picture we see $f=3$. A possible explanation is that one crosscap is superfluous as the red path shows.

The second picture has no superfluous crosscaps.

My question is, is there a simple way to identify superfluous crosscaps in more complicated situations? My only heuristic so far is that a "loop" of crosscaps has 1 superfluous crosscap.