Dominating Induced Matching in Some Subclasses of Bipartite Graphs

Panda, B. S.; Chaudhary, Juhi

doi:10.1007/978-3-030-11509-8_12

Cited by 2 publications

(2 citation statements)

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“…It turns out that the above problems are polynomial-time solvable on circular convex graphs and subclasses of (t, ∆)-tree convex graphs, but NP-complete for star convex graphs and comb convex graphs. In contrast, Panda and Chaudhary [35] proved that Dominating Induced Matching is not only polynomialtime solvable on circular convex and triad convex graphs, but also on star convex graphs. Nevertheless, we notice a common pattern: many dominating set, induced matching and graph transversal type of problems are polynomial-time solvable for (t, ∆)-tree convex graphs, for every t ≥ 1 and ∆ ≥ 3, and NP-complete for comb convex graphs, and thus for (∞, 3)-tree convex graphs, and star convex graphs, or equivalently, (1, ∞)-tree convex graphs.…”

Section: Mim-widthmentioning

confidence: 95%

“…It is worth noting that this complexity dichotomy does not hold for all LCVS problems; recall that Dominating Induced Matching is polynomial-time solvable on star convex graphs [35]. Theorems 1 and 2, combined with the result of [11], imply that this problem is also polynomial-time solvable on circular convex graphs and (t, ∆)-tree convex graphs for every t ≥ 1 and ∆ ≥ 3.…”

Section: Mim-widthmentioning

confidence: 99%

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Solving Problems on Generalized Convex Graphs via Mim-Width

Bonomo

Brettell

Munaro

et al. 2021

Lecture Notes in Computer Science

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A bipartite graph G = (A, B, E) is H-convex, for some family of graphs H, if there exists a graph H ∈ H with V (H) = A such that the set of neighbours in A of each b ∈ B induces a connected subgraph of H. Many NP-complete problems become polynomial-time solvable for H-convex graphs when H is the set of paths. In this case, the class of H-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of H-convex graphs where (i) H is the set of cycles, or (ii) H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can reprove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of H-convex graphs.Brettell and Paulusma received support from the Leverhulme Trust (RPG-2016-258). Bonomo received support from UBACyT (20020170100495BA and 20020160100095BA). Brettell also received support from a Rutherford Foundation Postdoctoral Fellowship, administered by the Royal Society Te Apārangi.

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Section: Mim-widthmentioning

confidence: 95%

Section: Mim-widthmentioning

confidence: 99%