Q2. What is the clique-width of a graph?
A class of graphs G has bounded clique-width if there is a constant c such that the clique-width of every graph in G is at most c; otherwise the clique-width of G is unbounded.
Q3. What is the bounded clique-width of a graph?
If G is a class of graphs and G′ is the class of graphs obtained from graphsin G by recursively deleting all vertices of degree 1, then G has bounded clique-width if and only if G′ has bounded clique-width [2,34].
Q4. What is the upper bound of the clique-width of graphs in G?
If the authors assume that Graph Isomorphism cannot be solved in polynomial time (a long-standing open problem), this would mean that for every class of graphs G on which the Graph Isomorphism problem is Graph Isomorphism-complete, the clique-width of graphs in G must be unbounded.
Q5. How many cliques can be removed from a graph?
by deleting at most two vertices of X, the authors obtain a graph which is a disjoint union of cliques and therefore has clique-width at most 2.
Q6. Why do the authors use graph operations that increase the upper bound of the clique-width?
This is because in their proofs the authors apply graph operations that exponentially increase the upper bound of the clique-width, which means that the bounds that could be obtained from their proofs would be very large and far from being tight.
Q7. What is the bounded cliquewidth of graphs?
Of the three classes for which the authors prove boundedness of clique-width in this paper, only the case of (2P1 + P2, 3P1 + P2)-free (and equivalently4 (2P1 + P2, 3P1 + P2)-free) graphs was previously known to be polynomial-time solvable [18].
Q8. What is the class of bipartite graphs with clique width?
The class of H-free bipartite graphs has bounded clique-width if and only if one of the following cases holds:• H = sP1 for some s ≥ 1 • H ⊆i K1,3 + 3P1 • H ⊆i K1,3 + P2 • H ⊆i P1 + S1,1,3 • H ⊆i S1,2,3.Lemma 5 ([25]).
Q9. What is the class of graphs with unbounded clique-width?
The class of (H1, H2)-free graphs has unbounded clique-width if(iv) H1 = 2P1 + P2 and H2 = P2 + P4.The authors prove statements (i)–(iv) of Theorem 1 in Sections 4–7, respectively.
Q10. How can the authors find clique-widths of unbounded graphs?
By modifying walls via graph operations that preserve unboundedness of clique-width, the authors are also able to present a new class of (H1, H2)-free graphs of unbounded clique-width.
Q11. What is the natural research direction for bounded graphs?
Another natural research direction is to determine whether the clique-width of (P1 + P4, H2)-free graphs is bounded for H2 = P2 +P3 (the clique-width is known to be unbounded for H2 ∈ {3P1 + P2, 2P1 + P3}).
Q12. What is the clique covering number of a graph G?
The clique covering number χ(G) of a graph G is the smallest number of (mutually vertex-disjoint) cliques such that every vertex of G belongs to exactly one clique.
Q13. What is the class of graphs that has bounded cliquewidth?
It is straightforward to verify that the class of H-free graphs has bounded cliquewidth if and only if H is an induced subgraph of the 4-vertex path P4 (see also [22]).
Q14. How many cases of graph Isomorphism have been proved?
Grohe and Schweitzer [26] recently proved that Graph Isomorphism is polynomial-time solvable on graphs of bounded clique-width.