Q2. What are the special vertices in an xi-gadget?
There are three special vertices in an xi-gadget: xi and xi, which the authors call literal vertices, and vxi which the authors call the head vertex of the xi-gadget.
Q3. What is the simplest way to determine whether to execute the while loop?
the total number of iterations of the while loop is bounded by the number of edges in G, so the authors see that the algorithm will terminate in polynomial time.
Q4. What is the maximum label assigned to an edge of a temporal graph?
An edge between two vertices u and v of G is denoted by uv, and in this case u and v are said to be adjacent in G. Given a temporal graph (G,λ), where G = (V,E), the maximum label assigned by λ to an edge of G, called the lifetime of (G,λ), is denoted by T (G,λ), or simply by T when no confusion arises.
Q5. What is the cutwidth of a graph?
The cutwidth of a graph G = (V,E) is the minimum integer c such that the vertices of G can be arranged in a linear order v1, . . . , vn, called a layout, such that for every i with 1 ≤ i < n the number of edges with one endpoint in v1, ..., vi and one in vi+1, ..., vn is at most c. Given a layout v1, v2, . . . , vn, the authors say that edges with one endpoint in v1, ..., vi and one in vi+1, ..., vn span vi, vi+1, and say that the maximum number of edges spanning a pair of consecutive vertices is the cutwidth of the layout.
Q6. What is the maximum temporal reachability of a subtree of G?
If (G,λ) is a temporal graph and v ∈ V (G), the authors say that T is a reachable subtree for v if T is a subtree of G, v ∈ V (T ) and, for all u ∈ V (T )\\{v}, there is a temporal path from v to u in (T, λ′), where λ′ is the restriction of λ to the edges of T .
Q7. What is the algorithm that returns a cutwidth approximation to Min TR Edge?
The authors claim that the following algorithm returns a c-approximation to Min TR Edge Deletion in polynomial time:1. Initialise E′ := ∅. 2. Initialise i := 0. 3. While (G,λ) has reachability greater than h:a.
Q8. What is the temporal reachability of a literal vertex?
Every clause vertex can reach only the corresponding literal vertices, their neighbours incident to the literal edges, and its own satellite vertex.
Q9. What is the deletion set for a graph?
At every iteration, the algorithm removes exactly h edges, while the optimum deletion set Eopt must remove at least one of these h edges.
Q10. what is the c-approximation of the cutwidth of the graph?
Add all edges that span vj , vj+1 to E′, and and update (G,λ)← (G,λ) \\ E′. c. Update i← j + 1 4. Return E′. JFor any fixed cutwidth c, using the layout generation algorithm given by Thilikos et al. [45] and the algorithm described above, the authors can give a cutwidth-approximation to Min TR Edge Deletion for graphs with cutwidth c.I Corollary 11.
Q11. How many edges can be removed from a temporal graph?
It is worth noting here that the (similarly-flavored) deletion problem for finding small separators in temporal graphs was studied recently, namely the problem of removing a small number of vertices from a given temporal graph such that two fixed vertices become temporally disconnected [26,51].3