# Spatial Graphs: Principles and Applications for Habitat Connectivity

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10021-007-9038-7

- Cite this article as:
- Fall, A., Fortin, M., Manseau, M. et al. Ecosystems (2007) 10: 448. doi:10.1007/s10021-007-9038-7

- 91 Citations
- 910 Views

## ABSTRACT

Well-founded methods to assess habitat connectivity are essential to inform land management decisions that include conservation and restoration goals. Indeed, to be able to develop a conservation plan that maintains animal movement through a fragmented landscape, spatial locations of habitat and paths among them need to be represented. Graph-based approaches have been proposed to determine paths among habitats at various scales and dispersal movement distances, and balance data requirements with information content. Conventional graphs, however, do not explicitly maintain geographic reference, reducing communication capacity and utility of other geo-spatial information. We present *spatial graphs* as a unifying theory for applying graph-based methods in a geographic context. Spatial graphs integrate a geometric reference system that ties patches and paths to specific spatial locations and spatial dimensions. Arguably, the complete graph, with paths between every pair of patches, may be one of the most relevant graphs from an ecosystem perspective, but it poses challenges to compute, process and visualize. We developed Minimum Planar Graphs as a spatial generalization of Delaunay triangulations to provide a reasonable approximation of complete graphs that facilitates visualization and comprehension of the network of connections across landscapes. If, as some authors have suggested, the minimum spanning tree identifies the connectivity “backbone” of a landscape, then the Minimum Planar Graph identifies the connectivity “network”. We applied spatial graphs, and in particular the Minimum Planar Graph, to analyze woodland caribou habitat in Manitoba, Canada to support the establishment of a national park.

### Key words

landscape modelingwoodland cariboupatchVoronoiDelaunay triangulationleast-cost path## INTRODUCTION

To make appropriate land management decisions (for example, species recovery plans, park establishment, harvest strategies), we need to understand animal responses to different habitat compositions (for example, quality) and configurations (for example, contiguous, fragmented, isolated) in terms of movement potential (Taylor and others 1993). Such species’ response to landscape features and patterns is known as *functional**connectivity* (With and others 1997; Tischendorf and Fahrig 2000). However, assessing functional connectivity directly is data intensive and requires lots of field experiments (Bélisle and others 2001; Brooks 2003; Bélisle 2005). Hence, it may be more feasible to assess the *structural connectivity* of habitat (that is, pattern analysis) for wide ranging species (Keitt and others 1997; Urban and Keitt 2001) or for species–habitat relations with long time lags (Manseau and others 2002), and to infer some aspects of functional connectivity. The challenge is to make a clear relationship between measures of structural and functional connectivity (With and others 1997; Brooks 2003).

A variety of methods have been proposed for assessing habitat connectivity (Schumaker 1996; Keitt and others 1997; With and others 1997; Calabrese and Fagan 2004) and fragmentation (Andrén 1994; Jaeger 2000). Landscape metrics that attempt to quantify connectivity using a single value are simple to implement and require relatively little data, but appear to be too simplistic to capture such a complex concept as structural connectivity, and results may be misleading or counter-intuitive (Gustafson 1998; Tischendorf 2001). At the other extreme, individual-based models implement detailed movement patterns (Fahrig and Merriam 1994; Schippers and others 1996; Goodwin and Fahrig 2002) and have the potential to yield key insights into functional connectivity, but tend to be more difficult to implement and have high data requirements to adequately parameterize for real species (Bélisle and Desrochers 2002).

Graph theory (Harary 1972) may provide a good compromise to bridge the goals of adequate correlation between functional and structural connectivity, and modest data requirements (Calabrese and Fagan 2004; Wagner and Fortin 2005). Marcot and Chinn (1982) proposed the application of graph theory for assessing the spatial configuration of wildlife habitat. Urban and Keitt (2001) developed some methods to study habitat connectivity using graphs and applied their approach to spotted owl (Keitt and others 1997) and prothonatory warbler habitat (Bunn and others 2000). Ricotta and others (2000) applied a variation of these methods to assess the structure of adjacent cover types, whereas Adriaensen and others (2003) explored least-cost paths across a landscape.

Our goal here is to lay the foundation for *spatial graph theory* and to highlight the importance of maintaining spatial reference of nodes and links. We place the insights from Urban and Keitt (2001) in this framework, and integrate spatial tessellation concepts (Okabe and others 2000). Tables are used to separate formal definitions to allow readers the choice of focusing on concepts or formal theory, and to facilitate ease of reference. Theoretical concepts are presented with illustrations so that the theory is accessible to modelers, researchers and practitioners.

We used spatial graphs to assess habitat connectivity for woodland caribou (*Rangifer tarandus caribou*) in central Manitoba, Canada to support the decision process surrounding the establishment of a national park. This habitat connectivity example utilized previous results in which parameters for spatial graph assessment were developed and validated for woodland caribou in other areas of Manitoba (O’Brien and others 2006). By examining the entire ecoregion that the national park is meant to represent, we were able to provide a broad geographic context in which the role of protected areas for conservation of caribou habitat and connections could be understood and communicated. Our aim is not to provide an exhaustive presentation of this analysis, but rather to demonstrate how the unique characteristics of spatial graphs helped transfer scientific information regarding connectivity to the decision process.

## METHODS

### Graph Theory Underlying Spatial Graphs

*nodes*and

*links*are usually called

*vertices*and

*edges*(Table 2). We use the former terms to avoid confusion with terms in common usage in landscape ecology. Graphs are usually drawn by representing nodes as dimensionless dots and links as lines connecting incident nodes, with both otherwise drawn in arbitrary locations. In spatial graphs, nodes are two-dimensional patches of habitat that represent specific locations with a given area and shape, and links connect nodes at specific points and follow specific georeferenced routes. A spatial graph can have links between node perimeters (that is, patch boundaries; Figure 1) or between interior node points (for example, patch centroids). Paths (a sequence of nodes connected by links in a linear way) and connected components (a subgraph of nodes that are jointly connected and all the links incident on these nodes) are core concepts for connectivity applications (Table 3).

Key Definitions of Graphs used for Spatial Graphs

• A (finite) |

• A link |

• A graph is |

• A |

• A |

Key Definitions of Graph Paths and Connected Components

• A |

• A |

• A graph is |

• A |

(i) |

(ii) For every link ( |

• An |

• A |

*nearest-neighbors*may result in more than one connected component. The minimum spanning tree (MST) consists of nearest neighbor links plus others to create a single connected component in which every link and non-leaf node is an articulation (for example, Figure 1A). An MST can be constructed by processing links in order of increasing weight; a link is added if it does not form a cycle with previously selected links, and is discarded otherwise. Hence, a link (

*n*

_{i},

*n*

_{j}) with weight

*w*is included if every other path from

*n*

_{i}to

*n*

_{j}has at least one link with a weight greater than

*w*. Because there is a unique path between every pair of nodes, an MST provides a concise way to represent the underlying “backbone” of connectivity in a landscape (Bunn and others 2000).

Key Definitions of Graph Types

• A |

• The nearest neighbor subgraph of a graph |

• A |

• A |

• A |

• A |

### Voronoi Diagrams and Delaunay Triangulations

Many applications in ecology need to map regions of influence surrounding point information (for example, sample plots, telemetry relocation points). A *tessellation* is a complete division of any finite portion of the Cartesian plane into non-overlapping regions. Given a set of input points, called *generators*, a *Voronoi diagram* is a tessellation into a set of convex polygons (called Voronoi, Dirichlet or Thiessen polygons), each of which represents the area closest to one of the generators (Okabe and others 2000). This “region of influence” concept generalizes to spatial patches, where Voronoi boundaries may no longer be straight lines due to the spatial nature of patches (for example, dashed lines in Figure 1A).

A *Delaunay triangulation* is formed by drawing a link between the nodes at the center of each adjacent pair of Voronoi polygons (Getis and Boots 1978). Each line is perpendicular to its corresponding polygon boundary segment, although it will not necessarily bisect it. This *triangulates* the nodes by dividing the minimum convex polygon of the nodes into non-overlapping triangles. We have generalized the Delaunay triangulation in spatial graphs, which we call the *minimum planar graph* (Figure 1A).

A Delaunay triangulation is a provably good approximation of a complete graph, yet contains far fewer links (Keil and Gutwin 1992). More specifically, the length of the shortest path between any pair of nodes in the Delaunay triangulation is not more than 2.43 times the Euclidean distance (Keil and Gutwin 1992). This property is important for ecological analysis, because organisms may potentially perceive all connections, but the complete graph (Figure 1B) is computationally expensive to construct and cumbersome to visualize.

### Spatial Domains

Definitions of Spatial Domains and Geometric Elements used for Spatial Graphs

• A (i) (ii) ( |

• A |

• A |

• A ( |

• The |

• A |

A *spatial cost function* is a spatially varying function *f*_{cost}(*p*) > 0 over every cell *p* where the value is interpreted as the cost in terms of one unit distance. Cost may reflect quantities such as movement energy or mortality risk, or, as in the case study, empirical observations of habitat selection. Euclidean distance is simply *f*_{cost}(*p*) = 1. Using a base cost of 1 in higher quality habitat, cost may be interpreted as the “effective” distance of a path (for example, a path with a cost of 2,000 units would have the same cost as 2 km through higher quality habitat). As defined, cost functions are independent of direction and can be represented as a surface (Figure 2A). Directed cost can be defined using a second point parameter (for example, cost may differ asymmetrically with elevation change). The cost of a line segment (*p*_{1}, *p*_{2}) is the sum of the cost values for each cell crossed times the length of the line crossing the cell. The cost along a line (*p*_{1}, *p*_{2},..., *p*_{k}) is the sum of the segment costs. Because *f*_{cost} increases monotonically along a line, it defines a metric space. That is, the cost of a line (*p*_{1}, *p*_{2},..., *p*_{k}) is less than the cost of any extension of the line (*p*_{1}, *p*_{2},..., *p*_{k}, *p*_{k+1}). A *least-cost path* between two points is a line with minimal cost. In Figure 2A, the least-cost path from the lower-left to the lower-right cell consists of three segments, with costs \( 2 \times 3{\sqrt 2 } \), 1.5 and \( {\sqrt {9 + 16} } \) for a total cost of about 15.0 over a distance of \( 3{\sqrt 2 } + 1 + 5 \approx 10.2 \) cell lengths. The straight-line cost is 62.5 units over 8 cell lengths. Implementations ought to minimize grid biases (for example, along cardinal axes or diagonals) present in some tools (for example, Adriaensen and others 2003).

### Spatial Graphs

Formal Definition of Spatial Graphs

• A |

(i) the nodes represent non-overlapping, contiguous regions (base patches) in |

(ii) a link |

(iii) the weight of link |

Special Types of Links that can be Defined for a Spatial Graph

• A |

• A |

• A |

• A |

• A |

*least-cost perimeter*links (that is, links along least-cost paths between patch perimeters, as in Figure 2B), some special types of spatial graphs can be generated from a set of nodes

*N*, spatial domain

*S*and spatial cost function

*f*

_{cost}(Table 4):

The

*minimum complete graph CG*= (*N*,*L*_{MI},*S*,*f*_{cost}) contains all minimum inter-node links, and is a complete graph of minimum cost.The

*minimum spanning tree MST*= (*N*,*L*_{MST},*S*,*f*_{cost}) has links that form a spanning tree of the nodes with minimum weight.The nearest-neighbor graph

*NNG*= (*N*,*L*_{NN},*S*,*f*_{cost}) contains all nearest-neighbor links. The nearest-neighbor graph is a subgraph of the MST (for example, Figure 1A), which is in turn a subgraph of the minimum complete graph (for example, Figure 1B).

### Minimum Planar Graphs

Planarity in conventional graphs is a property of the underlying graph, not a particular *drawing*, which is a mapping of the nodes and links, respectively, to points and lines in Cartesian space. A *planar graph* can be drawn on a two-dimensional plane with no crossing links (for example, Figure 1A). In conventional graph drawings, nodes can be placed arbitrarily and links can be any simple curve, and so planarity must be determined algorithmically (Reinhold and others 1977). A spatial graph and its drawing are inseparable, so planarity is easy to check. In addition to no crossing links, planarity of spatial graphs adds the further restriction that links cannot cross intermediate nodes (that is, links must be direct).

The *minimum planar graph* (MPG) is the spatial graph extension of the Delaunay triangulation. Voronoi boundaries are defined in spatial domains as lines of equal cost to two or more nodes. A *Delaunay**link* is a least-cost link *l* = ((*n*, *m*), (*p*_{n},..., *p*_{m})) between nodes *n* and *m* that share a Voronoi boundary in the spatial domain to which the points *p*_{n} and *p*_{m} are, respectively, the closest in nodes *n* and *m*. The MPG = (*N*, *L*_{MPG}, *S*, *f*_{cost}) generated from a set of nodes *N*, spatial domain *S* and spatial cost function *f*_{cost} contains the set of all direct Delaunay links. This is the maximal set of direct, non-crossing links where the resulting graph is of minimum weight. Indirect links indicate that nodes are co-linear in cost space (that is, three or more nodes fall on a line in cost space). An MPG has at most one link between each pair of nodes, and every link corresponds to a Voronoi boundary (but not every Voronoi boundary necessarily has an associated link). An MPG produces a generalized triangulation of the nodes where the apexes are patches not points and the sides may not be straight due to the cost function (Figure 2B). If the nodes are single points and the cost function is constant, the MPG is precisely a Delaunay triangulation, and Voronoi boundaries are conventional. With a cost function, if the nodes are single points, then the MPG is a triangulation in cost space. However, the finite spatial domain, the spatial nature of nodes and use of a cost function lead MPGs to have some unique properties.

*n*

_{4}and

*n*

_{6}are closer than

*n*

_{3}and

*n*

_{8}, precluding the minimum inter-node link between the latter. However, another link joining

*n*

_{3}and

*n*

_{8}satisfies the MPG definition. A node may also influence the location of an MPG link between other nodes (Figure 4). To maintain the

*empty-circle*property of Delaunay triangulations (that is, for any link, there exists a circle with the endpoints on the perimeter that contains no other points), the start and end points of an MPG link are the points closest to the center of the minimum empty circle that touches the two nodes. A nearby node may displace an MPG link even if it does not block the minimum inter-node path (for example, Figure 4b). Hence, the MPG is not necessarily a subgraph of the minimum complete graph.

The MPG may not be a complete triangulation due to the finite spatial domain or the cost function. A Voronoi boundary may lie entirely outside of the study area, so no link is made (for example, between nodes *n*_{4} and *n*_{5} in Figure 1A). Landscapes of interest are usually embedded within a larger region, so it is hard to justify links that may be precluded in a larger area. An MPG must also respect the metric cost space, in particular the *triangle inequality*, which states that the cost of a link from *n*_{1} to *n*_{2} cannot exceed the cost from *n*_{1} to *n*_{3} plus the cost from *n*_{3} to *n*_{2}.

As a generalization of a Delaunay triangulation, the MPG is likely a good approximation of the complete graph (although this has not been formally proven). The minimum spanning tree is a subgraph of the MPG. The MPG adds further links to the MST to join nodes that are closest without violating planarity. Figure 1 illustrates how an MPG can improve visualization compared to a complete graph. Even with only eight nodes, the complete graph is cluttered with many links crossing nodes and each other, making it difficult to understand key patterns. The MPG has fewer links, but maintains the network of connections in a form that is comprehensible.

### Spatial Graph Analysis

Patches are often defined as contiguous regions that are more internally similar than with surrounding areas (Pickett and White 1985). In a grid-based representation, a definition based on cell adjacency is a common basis of many patch-based analyses (for example, McGarigal and Marks 1995; Gustafson 1998). However, this approach is problematic because a patch should be defined based on a species’ perception of a landscape, not on data resolution. Depending on the resolution, there will be more smaller, or fewer larger patches for the same landscape (Jelenski and Wu 1996). An advantage of graph-based analysis is that we do not need to predefine the scale of patches. Patches cannot be identified at resolutions finer than the input data, but they can be identified at coarser scales, leading to a multi-scale analysis of patch patterns that forms a basis of the approach developed by Urban and Keitt (2001).

We define contiguous areas at the base resolution as nodes or *base patches*, but let the general concept of a patch have a scaled definition: at a given *threshold weight w*, we assume that movement is impeded along links with weights greater than *w* by removing all links from *G* with weights greater than *w*. Formally, a *scaled subgraph**G*^{w} of a graph *G* contains all the nodes of *G* and all the links with weights less than or equal to the threshold weight *w*. The resulting subgraph is comprised of a set of connected components that we call *clusters*. At the finest scale, where *w* = 0, clusters are simply the base patches. At coarser scales, clusters consist of groups of linked nodes. For scales at or above the maximum weight link in the MST of *G*, there is just a single cluster. Thus, if a particular species perceives the landscape at scale *w*, then the patches can be defined using the clusters of scaled graph *G*^{w}. The scaled definition of a patch as a cluster allows scaling of patch analysis metrics. Hence, for a given scaled spatial graph *G*^{w} we can define the number of clusters, mean cluster size, largest cluster size, and so on.

*Expected cluster size*is the size of cluster in which a randomly selected cell is expected to reside at a given threshold

*w*:

*n*is the number of cells in the landscape, nc

^{w}is the number of clusters at threshold

*w*,

*n*

_{j}

^{w}is the number of cells in cluster

*j*, and

*cluster*(

*w,i*) is the cluster in which cell

*i*resides. When

*w*= 0, ECS is equivalent to area-weighted mean patch size and is related to effective mesh size (Jaeger 2000). ECS increases monotonically with increasing thresholds. By plotting the independent threshold variable

*w*on the

*x*-axis and ECS

^{w}on the

*y*-axis, one can identify thresholds or ranges of thresholds over which rapid increases in connectivity can be identified. These

*critical thresholds*(called

*critical scales*in Keitt and others 1997) can be used to identify scales at which habitat becomes connected. Comparison with movement capabilities of the species of interest (at a given behavioral scale or timeframe, such as foraging, migration, genetic flow) can be used to assess the degree to which a landscape is likely perceived as connected. The ability to apply a posteriori estimates of movement capabilities (as opposed to approaches that require a priori estimates) is a key strength because the uncertainty surrounding such estimates is likely higher than uncertainty around structural landscape patterns, and so this approach facilitates error analysis and sensitivity analysis of assumptions regarding organism movement.

## A CASE STUDY: SPATIAL GRAPHSTO IDENTIFY CORE WOODLAND CARIBOU HABITATAND CONSERVATION NEEDS

Woodland caribou (*Rangifer tarandus caribou*) is a boreal species of ungulate that is currently designated as threatened by the Committee on the Status of Endangered Wildlife in Canada (COSEWIC 2001). The protection of woodland caribou became an issue surrounding the establishment of a national park in central Manitoba as the proposed boundaries failed to include sufficient habitat to ensure the long-term viability of the residing herd (Manseau and others 2001). Because this herd is at the southern limit of the species range, it is important to delineate park boundaries that capture a significant amount of high quality habitat as well as maintain connectivity to the north to allow for long-range movements and population exchanges. We applied a spatial graph analysis to assess connectivity patterns and the spatial distribution of habitat, and to communicate findings to the decision process surrounding park establishment. Of particular interest were issues related to ecologically defined park boundaries, the role of the proposed park in conserving habitat and connectivity for this wide-ranging species, and broad scale management of this species in the ecoregion context.

### Case Study Methods

The landscape consisting of the entire Manitoba Lowlands Natural Region (Figure 5) is just under 16 million hectares, which we represented using a resolution of 1 ha/cell (100 m × 100 m). We generated a habitat classification and a cost surface using Manitoba Forest Resource Inventory information and parameters from an approach we developed and validated elsewhere in Manitoba (O’Brien and others 2006).

A key habitat required by woodland caribou in many parts of the boreal region is late seral jack pine (*Pinus banksiana* Lamb.) forest that supports a large ground lichen biomass (Darby and Pruitt 1984; Brown and Theberge 1990; Schaefer 1996). The caribou habitat model classifies each cell in the study area into high quality (jack pine stands 60 years and older), medium quality (treed muskeg, and mature conifer uplands over 71 years), low quality (mature conifer lowlands 60 years and older, and conifer-dominated mixedwood uplands 46–71 years), very low quality (young forest, and hardwood-dominated mixedwood), recent burns, water, communities, roads and trails. We defined the base patches as *high quality habitat* in contiguous areas of at least 10 ha, which resulted in 4,574 patches.

Impedance values for the cost surface were defined in terms of the probability of using habitat of varying quality, relative to high quality habitat. We developed a resource selection function using logistic regression (Manly and others 2002) to compute an impedance value for each habitat class based on analysis of GPS relocation data for the Owl Lake woodland caribou herd in southeastern Manitoba, which resulted in 1.00 for high quality habitat (that is, cost increases linearly with distance), 1.01 for medium quality, 1.73 for low quality, 3.30 for recent burns, 3.54 for very low quality and 3.60 for water (O’Brien and others 2006). A value of 5.00 was used for linear road and transmission line features (suitably represented to avoid “cracks”) based on expert opinion, as no suitable empirical data was available.

We constructed the MPG using the cost surface, and assessed the expected cluster size across a series of increasing cost thresholds to identify scales at which caribou habitat increases in connectivity. In addition, we compared the error of paths between nodes through the MPG with the minimum complete graph (for feasibility of this comparison, we used 200 m × 200 m resolution, which resulted in 2,744 patches). Spatial graph models were implemented in Spatially Explicit Landscape Event Simulator (SELES; Fall and Fall 2001; http://www.seles.info) and are available upon request to the authors.

### Case Study Results

At 200 m resolution, the complete graph had 3,763,396 links, whereas the MPG had only 7,707. The average path error (measured as the percent by which paths through the MPG were longer than in the complete graph) was 5.6% for paths between pairs of patches, and 4.0% for paths between pairs of habitat cells from different patches. These errors do not affect the graph thresholding analysis, as this is driven by the minimum spanning tree.

### Case Study Discussion

A spatial representation of landscape connectivity from the woodland caribou perspective clearly highlighted management issues that will be central to this future national park. Connectivity between high quality habitats is limited by the large lakes, constraining animal movements to a north–south axis between Lake Winnipeg and Lake Winnipegosis. Furthermore, the Cedar Lake reservoir, recently built highways and communities present additional barriers, weakening the north–south axis to an island-hopping route across Cedar Lake or through the outskirts of Grand Rapids. This suggests that the range of the Long Point herd is highly spatially constrained (a finding corroborated by recent genetic analysis, Pither and others 2005), with only a few links, as identified by the MPG, potentially serving as movement corridors. By clarifying these major landscape fragmentation issues and conservation challenges for the long-term viability of this herd, our analysis suggested that the proposed park boundaries did not adequately capture habitat clusters and connectivity for caribou.

Spatially mapping the MPG links provided an effective means of communication with all parties involved in the park establishment process, and helped lead to a reconfiguration of the park boundaries. The boundaries of the proposed protected area were re-oriented along a north–south axis, both conceptually and for ecological reasons. The Long Point area was modified to encompass more habitat clusters primarily to the south and contiguity with a provincial park reserve also to the south. Because boundaries couldn’t include important areas to the north of Long Point (because of industrial activities and communities), our analysis helped to inform discussions on protecting stepping stones, corridors and clusters of habitat to the north by provincial jurisdictions (for example, through land use and forest management plans). Although it is evident that a single protected area (even as large as the proposed national park) is not sufficient to ensure the viability of this (and other) wide-ranging species over long time frames, striving to define ecological boundaries and identify future conservation challenges at the onset remains important.

## DISCUSSION

Spatial graph theory synthesizes previous work in the area of applying graph-based methods to habitat connectivity analysis. This theory formally integrates conventional graphs with spatial referencing, and makes extensions to include spatial dimensions of patches and spatial pathways of links. We highlighted important considerations for designing algorithms to “generate” link locations and costs, and for interpreting results. Maintaining spatial referencing is essential for effective communication, analysis and management decision-support.

The MPG is a spatial generalization of the Delaunay triangulation, which provides a good approximation of the complete graph. Complete graphs are ecologically relevant from the perspective of optimal movement through a landscape, but are generally too large for efficient extraction, analysis and visualization. In the case study, we demonstrated that the mean deviation of paths between patches through the MPG was only 5.6% longer than in the complete graph, with only 0.2% as many links. Hence, the MPG provides a good underlying graph for spatial multi-scale connectivity analysis. If the MST identifies the connectivity “backbone” of a landscape (Bunn and others 2000), then the MPG identifies the connectivity “network”. Backbones are fragile structures, and a single broken link disconnects an MST. The MPG captures redundancy in a system and alternative paths between nodes to give a more complete system view, while retaining communication parsimony and computational feasibility.

We demonstrated the utility of applying spatial graphs to present an ecologically meaningful landscape perspective of a study region to biologists and managers. By identifying key features critical for the persistence of landscape scale ecological processes, the graph-based analyses helped to delineate the boundaries of a national park, and to expose current and future management challenges. The analyses revealed the importance of stepping-stone islands that could be restored to re-establish connectivity between the southern and northern parts of the landscape. The spatial visual representation conveyed the importance of the locations of forestry activities and linear features in diminishing the functional role of habitat patches. A functional representation of the area highlighted both opportunities and challenges for conservation, and the necessity to work with other sectors of government, communities and industries. Spatial graphs, and in particular the MPG, were shown to be very valuable in communicating the network of connections in a comprehensible and comprehensive manner, which was necessary for decision-support.

In addition to connectivity analysis, spatial graphs have a diverse range of potential ecological applications. Many pattern metrics are based directly on grid resolution (for example, McGarigal and Marks 1995; Qi and Wu 1996; Cain and others 1997). Spatial graphs provide a method to distinguish between representation scale (that is, grid resolution) and ecological scale (that is, spatial grain relevant to ecological processes of interest), and so can generalize such landscape metrics by supporting multi-scale analyses that reduce grid artifacts. Spatial graphs can be used in conjunction with meta-population analysis and population viability analysis to quantify the spatial attributes of species habitat requirements and to identify the strength of linkages within and between populations (Sutherland and others 2006). Structural and genetic connectivity can be compared to estimate the scale at which a species is genetically adapted (Brooks 2003), and the degree to which these deviate can provide insights into the likely pressures facing a species in a managed landscape. By providing a powerful means for communicating the network of connections across a landscape, in particular alternatives and options for maintaining or restoring connectivity, Minimum Planar Graphs could be a practical tool to support ecosystem-based management planning.

We have presented a foundation for spatial graphs, upon which further theory and methods can be built. Expanding on the concept of multiple pathways between patches, methods could be developed to delineate and analyze corridors as spatial objects. Linkages with other planning tools (for example, dynamic simulation models or timber planning models) would enhance the ability to use graphs for decision support for conservation and recovery.

## ACKNOWLEDGEMENTS

This research was funded by Parks Canada Species at Risk Recovery Action and Education Fund, a program supported by the National Strategy for the Protection of Species at Risk and Parks Canada Western Canada Service Research Fund as well as GEIODE Strategic Initiative Program. Participating partners of this project were Manitoba Hydro, Manitoba Conservation, Tolko Ltd. and Natural Resources Institute of the University of Manitoba. We thank Jennifer Keeney for collating data and doing the GIS and mapping work. We also thank Richard Pither, Pasi Reunanen, Patrick James, Glenn Sutherland, Doug Steventon and two anonymous reviewers for helpful comments.