Ptolemaic and Planar Cover-incomparability Graphs

Art Discrete Appl. Math.

2022

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs arising from posets. A few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.

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Section: Introductionmentioning

confidence: 99%

“…C-I graphs are identified among the planar graphs and chordal graphs along with new characterizations of Ptolemaic graphs, respectively in [4] and [3]. It is also interesting to note that every C-I graph has a Ptolemaic C-I graph as a spanning subgraph [4]. In a most recent paper, C-I graphs are studied among comparability graphs [2].…”

Section: Introductionmentioning

confidence: 99%

Composition and product of cover-incomparability graphs

Art Discrete Appl. Math.

2022

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“…Such C-I graphs studied include the family of split graphs, block graphs [7], cographs [8], Ptolemaic graphs [22], distance hereditary graphs [22], and k-trees [23]. The C-I graphs were identified among the planar and chordal graphs along with new characterizations of the Ptolemaic graphs, respectively, in [4] and [3]. It is also interesting to note that every C-I graph has a Ptolemaic C-I graph as a spanning subgraph [4].…”

Section: Introductionmentioning

confidence: 99%

“…The C-I graphs were identified among the planar and chordal graphs along with new characterizations of the Ptolemaic graphs, respectively, in [4] and [3]. It is also interesting to note that every C-I graph has a Ptolemaic C-I graph as a spanning subgraph [4]. C-I graphs are also studied among the comparability graphs [2].…”

Section: Introductionmentioning

confidence: 99%

Ptolemaic and Chordal Cover-Incomparability Graphs

2021

Order

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Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset P = (V, ≤) with vertex set V , and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. The recognition of the C-I graphs is known to be NP-complete (Maxová et al., Order 26(3), 229-236(2009)). In this paper, we prove that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs. A similar result is obtained for cographs as well. Using the structural results of these graphs, we derive linear time recognition algorithms for chordal graphs and cographs, which are C-I graphs.

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Comparability Graphs Among Cover-Incomparability Graphs

Algorithms and Discrete Applied Mathematics

2022