# Algorithm Based on Morphological Operators for Shortness Path Planning
## 2 Morphological Search Framework
### Preliminary Concepts  
Consider a set of nodes \(N=\{n_{1},\ldots,n_{k}\}\) and a set of edges \(E=\{e_{1},\ldots,e_{l}\}\). A graph consists of \(N\times E\), which is denoted by \(G=(N,E)\), and the definition is represented by an enumerated set \(G=\{(n,e)|n\in N,e\in E\}\). An arbitrary graph \(G\) represents the coding of a scenario [10; 39]. The \(G\) graph represents a net topology of connections, where the nodes and edges denote the available places and steps. A particular lattice is denoted by a two-tuple \(O=(L,\leq_{G})\), where \(L\subseteq 2^{G}\) is an arbitrary subset and \(\leq_{G}\) is a partial-order relation in \(2^{G}\). The partial-order relation \(\leq_{G}\) considers an arbitrary \(x_{0}\subseteq G\) reference and denotes the allowed movements and directions in the navigation of \(L\). In our case, we consider a lattice that covers all subsets of adjacent nodes that cover the graph space starting from the \(x_{0}\subseteq G\) reference. The order by which adjacent nodes are covered, resulting in subgraphs, depends on \(L\), where the reference \(x_{0}\) defines the order relation. In other words, the supreme element of the lattice is the entire graph \(L\). The infimum represents the element \(e\subseteq G\) such that there is no other \(e^{\prime}\) that belongs to \((e^{\prime},e)\in\leq_{G}\); for simplicity, it is expressed as \(\textbf{sup}\leq_{G}\) and \(\textbf{inf}\leq_{G}\), respectively. As a complement, a path lattice in \(L\) is a set of graphs of the step incremental connected graph starting from the \(x_{0}\subseteq G\) reference and ending at references \(x_{k}\subseteq G\). The path is covered by all steps over \(\leq_{G}\) to reach the \(x_{k}\) graph. Regarding the order relation \(\leq_{G}\), the reference \(x_{0}\) becomes a subset of \(x_{k}\). The path lattice is expressed as an ordered collection \(p=\{p_{1},\ldots,p_{k}\}\) and describes the incremental step cover path resulting from adding the graph neighborhood under a connectivity criterion denoted by the connection order in the graph space. Two subgraphs \(x_{i}\in p\), \(x_{i+1}\in p\) become adjacent under the \(\leq_{G}\) whenever it is found that the relation \((x_{i},x_{i+1})\in\leq_{G}\) belongs to the order relation. For simplicity, the path can be expressed as the number of steps from a reference \(x_{0}\) to the ending \(x_{k}\) as \(\underbrace{x_{0}\leq_{G}\ldots\leq_{G}x_{k}}_{k\text{-times}}\). These steps represent all intermediate adjacent steps in \(p\). The lattice denotes the directionality and availability transitions over the allowed subgraph in \(G\). The graph topology and the order relation \(\leq_{G}\) depend on the context of each scenario. Using morphological operators in the graph space \(G\), we can define a partial order \(\leq_{G}\) criterion, establishing the incremental criterion from the boundaries of the current subgraph. As mentioned above, mathematical morphology (MM) introduces two basic operators known as _dilation_ and _erosion_. Both use a graph structure to navigate the graph neighborhood, called a structural element. The \(\lambda\) parameter denotes the structural element representing the step-up/-down reached/dismissed nodes from the current subgraph across the basic MM operators. The operator of _dilation_ (\(\delta\)) is an increasing function that denotes a step forward \(x_{0}\leq_{G}x_{1}\) following increasing coverage for a given lattice \((L,\leq_{G})\) with a unitary step \(\lambda\). The dilation on a complete lattice is expressed as follows:  
\[\delta_{\lambda}(A)=\bigcup_{b\in\lambda}a_{b} \tag{1}\]  
where \(A\in L\) and all nodes \(a\in A\); \(a_{b}\) denotes the addition of the \(b\) isomorphic node referring to \(a\) if and only if \(a_{b}\in G\), from a given neighborhood \(\lambda\), where \(\lambda\) is a graph that denotes the homomorphic structure and \(G\) represents the graph space.