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Consider $G[X]$, where $G$ is a connected, isthmus-less and labelled graph,and $X$ is the edge-set or the vertex-set of the graph $G$. A Graphical SearchProblem (GSP), denoted $\Pi(G[X],\gamma)$, consists of finding $Y$, where $Y\subseteq X$ and $Y$ satisfies the predicate $\gamma$ in $G$. The subset $Y$ isa solution of the problem $\Pi(G[X],\gamma)$. A sub-solution of$\Pi(G[X],\gamma)$ is a subset $Y'$ such that $Y'$ is not a solution of$\Pi(G[X],\gamma)$, but $Y'$ is a solution of the problem $\Pi(H[X'],\gamma)$,where $X' \subset X$ and $H[X']$ is a contraction-minor of $G[X]$. Solutionsand sub-solutions are the feasible sets of $\Pi(G[X],\gamma)$.  Let $\mathfrak{I}$ be the family of all the feasible sets of$\Pi(G[X],\gamma)$. A Hyper-greedoid is a set system $(X, \mathfrak{I})$satisfying the following axioms.  A1: Accessibility: if $I \in \mathfrak{I}$, there is an element $x \in I$such that $I-x \in \mathfrak{I}$  A2: Augmentability: If $I$ is a sub-solution, there is a polynomial timefunction $\kappa: \mathfrak{I} \rightarrow \mathfrak{I}$ and there is a element$x \in X-\kappa(I)$ such that $\kappa(I) \cup x \in \mathfrak{I}$. That is,every sub-solution can be augmented using a polynomial time algorithm akin toEdmond Augmenting Path Algorithm.  Given a graph $G$, the GSP MISP consists of finding an independent set ofvertices of $G$. MISP satisfies axioms A1 and A2. Using the P-completeness ofthe Decision Problem associated to MISP, we prove that every GSP that satisfiesA1 is solvable in Polynomial Time if and only if it satisfies A2. On the otherhand, let HCP be the GSP that consists of finding a Hamiltonian cycle of thegraph $G$. HCP satisfies A1, but does not satisfies A2. Since the DecisionProblem associated with HCP is NP-Complete, we get $P \not = NP$.

There is a Hyper-Greedoid lurking behind every Graphical Accessible Computational Search Problem solvable in Polynomial Time: $P \not= NP$