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Graphbots: Mobility in Discrete Spaces

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Sharon Lahav

Graphbots: Mobility in Discrete Spaces

- Move beyond robots with simple geometries (polygonal structure).
- Move beyond simple spaces (planar region containing polygonal obstacles).
- Teams of robots that operate in discrete spaces like graphs.
- And that have discrete geometries represented by subgraphs i.e they are maintaining a formation!

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- A connected graph G - “the graph space”
- A connected subgraph of G H – “a cooperating team of robots”

- We represent the members of the team by single nodes.

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- A movement or motion of H from S to T is defined by a sequence of subgraphs: S=H0, H1,…,Hk=T all isomorphic to H.
- The structure must be preserved when the team moves.

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- Find conditions for G to satisfy a free movement of a given subgraph H in G.
- Establish the complexity of finding a motion with the fewest local displacements from S to T (if one exists).

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- A “Tick” is modeled by a two vertex graph linked by a single edge.

- Theorem 2.1 - A tick can move freely in any connected graph.

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- A “Scorpion” is modeled by a threevertex graph linked by two edges.
- The degree2 vertex is the “body”.
- The degree1 vertices are the “feet”.

- Theorem 2.2- A scorpion can move freely in G iff G does not contain a vertex v with two neighbors of degree 1.

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b

- If G has such a vertex v, we can place the scorpion with b on v, and the f 's on the neighbors vi of v.
- Any movement of the feet requires that both must move to v. If we move one foot to v, then b must leave v the other foot will not be adjacent to b's new location.

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G

S

T

- There is no such vertex v.
- S-{v1,v2,v3} , T-{u1,u2,u3}
- If there is a path joining a foot of the scorpion in the initial S location to a foot at the final T location, without passing through u2 and v2 , then the scorpion can “creep” along this path .

f b f

f b f

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u2

G

u2

v1

v3

S

T

u3

u1

u3

u1

v2

u0

u0

- The only path from S to T goes through either u2 or v2 (or both):

- The degree of u1 is not 1 and it has a neighbor u0 (other than u2)
- f goes from u1 to u0 , b goes to u1 and f goes from u3 to u2 . Now the scorpion can creep along the path to T.

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To corner a scorpion

you have to completely

immobilize it at its

starting location!

b

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- A single vertex in a connected graph whose deletion disconnects the graph is called a cut vertex.
- Biconnected Graphs - A graph with no cut vertices is called biconnected.
- - Connected Graphs - A graph is said to be Connected if the deletion of any subset of -1 vertices leaves the graph connected.

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- A Chordal Graph is a graph in which each cycle of length at least 4 has a Chord.
- A Chord is an edge that connects two vertices that are not adjacent in the cycle.

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- A perfect elimination ordering(peo) is a numbering of the vertices from {1,…,n} such that for each i, the higher numbered neighbors of vertex i form a clique.
- A peo is represented by a sequence of vertices.
- Theorem 2.3 (Fulkerson and Gross) A graph G has a peo iff G is chordal.

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- A “Trilobite” is modeled by a threevertex graph linked by three edges. (a clique of size three).
- Theorem 2.4- A Trilobite can move freely in a biconnected chordal graph, that has at least three vertices.

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chordality is sufficient,

but not necessary.

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- A spider is modeled by a (k+1)vertex graph having a central vertex denoting its “body” linked by edges to vertices f1,…,fk representing its “feet”.
- K=1 is a “Tick”
- K=2 is a “Scorpion”
- Theorem 2.6 A klegged Spider can move freely in a (K-1) Connected Chordal Graph.

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- We can move a threelegged spider in a biconnected chordal graph, this follows from the theorem 2.6 with k = 3.
- A stronger result yet!Theorem 2.10 A fourlegged spider can also move freely in a biconnected chordal graph.

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A fivelegged spider

cannot move freely

in a biconnected

chordal graph!

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- If G has n vertices and H has l vertices. There are at most O(nl) possible locations of H in G.
- G’ = (V’,E’) in which each vertex corresponds to a possible valid location of H in G.
- There is an edge in E’ between u,vV’ if there is a local displacement between u and v of H.

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- G’ can be constructed in polynomial time in the size of any fixed graph H.
- By finding the shortest path in G’ from S to T, we can determine the motion with the least number of local displacements in polynomial time.

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- Clique(G,k) : is the problem of checking if the graph G contains a clique of size k. This problem is known to be NPcomplete!
- We reduced the clique problem to checking to see if there exists a motion that moves H = K 2k from a start location to a target location.

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- By constructing a new graph G’ =(V’,E’) V’=V(x1,…,x2k)(y2k+1,…,y4k). E’=EExEyE’’ Ex=[(xi,xj)|1ij2k] Ey=[(yi,yj)|2k+1ij 4k] E’’=[(v,xi),(v,yj)| vV,1i2k ,2k+1j4k]

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סוף

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