So we have \( g ( P _ { 3 , o _ { 1 } } ^ { \rightarrow } ) \neq g ( P _ { 3 , o _ { 2 } } ^ { \rightarrow } ) \).

Considering the partial graph \( G - P _ { 3 } \), let \( g ( ( G - P _ { 3 } ) _ { o _ { i } } ^ { \rightarrow } = t \) for a specific orientation \( o _ { i } \) and a predator-prey strategy \( s _ { k } \).

For the path \( P _ { 3 } = v _ { 1 } v _ { 2 } v _ { 3 } \) consider the graph \( G ^ { \rightarrow } \) with the predator-prey strategy \( s _ { k } ^ { * } = \)\( ( s _ { 1 } , s _ { k } ) \) and the orientation \( o _ { i } ^ { * } = \{ ( v _ { 1 } , v _ { 2 } ) , ( v _ { 2 } , v _ { 3 } ) , o _ { i } \} \). Clearly arcs to and from \( P _ { 3 } ^ { \rightarrow } \) and the rest of \( G ^ { \rightarrow } \) may exist such that upon applying \( s _ { k } ^ { * } \) we have \( g ( ( G \mathrm { ~ - ~ } P _ { 3 } ) _ { o i } ^ { \rightarrow } ) \mathrm { ~ - ~ } 4 \mathrm { ~ = ~ } t \mathrm { ~ - ~ } 4 \mathrm { ~ \leq ~ } \)\( g ( G ^ { \rightarrow } ) _ { o _ { i } ^ { * } } \leq t + 2 = g ( ( G - P _ { 3 } ) _ { o _ { i } } ^ { \rightarrow } ) + 2 \).

For the path \( P _ { 3 } \, = \, v _ { 2 } v _ { 1 } v _ { 3 } \) consider the graph \( G ^ { \rightarrow } \) with the predator-prey strategy \( s _ { k } ^ { * } = \)\( ( s _ { 1 } , s _ { k } ) \) and the orientation \( o _ { j } ^ { * } = \{ ( v _ { 2 } , v _ { 1 } ) , ( v _ { 1 } , v _ { 3 } ) , o _ { j } \} \). Clearly arcs to and from \( P _ { 3 } ^ { \rightarrow } \) and the rest of \( G ^ { \rightarrow } \) may exist such that upon applying \( s _ { k } ^ { * } \) we have \( g ( ( G - P _ { 3 } ) _ { o _ { i } } ^ { \rightarrow } ) - 8 \, = \, t - 8 \, \leq \)\( g ( G ^ { \rightarrow } ) _ { o _ { j } ^ { * } } \leq t + 4 = g ( ( G - P _ { 3 } ) _ { o _ { j } } ^ { \rightarrow } ) + 4 \).

Since the set of arcs between the two directed paths and the rest of \( G ^ { \rightarrow } \) must specifically remain the same, it follows that:

\[ \begin{array} { c } { { g ( ( G - P _ { 3 } ) _ { o _ { i } } ^ { \rightarrow } ) - 4 = t - 4 \leq g ( G ^ { \rightarrow } ) _ { o _ { i } ^ { * } } \leq t + 2 = g ( ( G - P _ { 3 } ) _ { o _ { i } } ^ { \rightarrow } ) + 2 \neq } } \\ { { g ( ( G - P _ { 3 } ) _ { o _ { j } } ^ { \rightarrow } ) - 8 = t - 8 \leq g ( G ^ { \rightarrow } ) _ { o _ { j } ^ { * } } \leq t + 4 = g ( ( G - P _ { 3 } ) _ { o _ { j } } ^ { \rightarrow } ) + 4 . } } \end{array} \]

Therefore, in general the result that there exist at least two different orientations and at

least two different indices of G, such that \( g ( G _ { o _ { i } } ^ { \rightarrow } ) \neq g ( G _ { o _ { j } } ^ { \rightarrow } ) \), follows.

Proposition 2.7. If a cycle \( C _ { n } , n \geq 3 \) and any specific orientation thereof say, \( C _ { n } ^ { \rightarrow } \) is ex-tended to \( C _ { n + 1 } ^ { \rightarrow } \) the residual population over all possible predator-prey strategies applicable to \( C _ { n + 1 } ^ { \rightarrow } \) is given by:

\[ r _ { s _ { k } ^ { * } } ( C _ { n + 1 } ^ { \rightarrow } ) = r _ { s _ { k } } ( C _ { n } ^ { \rightarrow } ) + ( n - 1 ) , \]

and \( s _ { k } ^ { * } \) is the minimal deviation from \( s _ { k } \) to accommodate arcing to or from \( v _ { n + 1 } \).

Proof. Consider \( C _ { n + 1 } \). We begin by considering any specific orientation of \( C _ { n } \) and denote it \( C _ { n } ^ { \rightarrow } \). Clearly the extension from \( C _ { n } ^ { \rightarrow } \) to \( C _ { n + 1 } ^ { \rightarrow } \) is made possible by squeesing \( v _ { n + 1 } \), (arcing) inbetween two vertices of \( C _ { n } ^ { \rightarrow } \), say \( v _ { s } , v _ { t } \) with 1 \( \leq s , t \leq n \).

Without loss of generality consider the arc \( ( v _ { s } , v _ { t } ) \) in \( C _ { n } ^ { \rightarrow } \). Begin by letting \( v _ { s } \) prey on \( v _ { t } \) by predatoring along the arc \( ( v _ { s } , v _ { t } ) \). After this first step of the Grog algorithm the rest of the application applies to a path \( P _ { n * } ^ { \rightarrow } \) with end vertices \( v _ { s } \) and \( \boldsymbol { v } _ { t } \) having \( \rho ( v _ { s } ) = s - 1 \) and \( \rho ( v _ { t } ) = t - 1 \). After applying \( s _ { k } \) to this path the value \( r _ { s _ { k } } ( C _ { n } ^ { \rightarrow } ) \) is obtained.

We now squeese \( v _ { n + 1 } \) inbetween \( v _ { s } , v _ { t } \) and for any one of the four possible orientation between \( v _ { s } , v _ { n + 1 } , v _ { t } \) we have that if the Grog algorithm is applied to path \( P _ { 3 } ^ { \rightarrow } \), \( V ( P _ { 3 } ) = \)\( \{ v _ { s } , v _ { n + 1 } , v _ { t } \} \) we obtain \( \rho ( v _ { s } ) = s - 1 \), \( \rho ( v _ { n + 1 } ) = n - 1 \), \( \rho ( v _ { t } ) = t - 1 \).

Furthering with the Grog algorithm we are left with exactly the path \( P _ { n * } ^ { \rightarrow } \) mentioned above. Hence the result: