Irredundancy in circular arc graphs

Martin Charles Golumbic; Renu C. Laskar

doi:10.1016/0166-218X(93)90223-B

Irredundancy in circular arc graphs

Research output: Contribution to journal › Article › peer-review

Abstract

A set of vertices X is called irredundant if for every x in X the closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 2β^* or 2β^*+1 (depending on parity) where β^* is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K₁'s and K₂'s. Similar results are proven for bipartite graphs.

Original language	English
Pages (from-to)	79-89
Number of pages	11
Journal	Discrete Applied Mathematics
Volume	44
Issue number	1-3
DOIs	https://doi.org/10.1016/0166-218X(93)90223-B
State	Published - 19 Jul 1993
Externally published	Yes

Bibliographical note

Funding Information:
Correspondence to; Professor MC. Golumbic, Department of Mathematics and Computer Science, Bar-Ban University, Ramat Gan, Israel. * While on leave of absence from the IBM Israel Science and Technology Center, Technion City, Haifa, Israel. ** This work was supported in part by contract NOO014-86-K-0693 from the Office of Naval Research.

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1016/0166-218X(93)90223-B

Cite this

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title = "Irredundancy in circular arc graphs",

abstract = "A set of vertices X is called irredundant if for every x in X the closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 2β* or 2β*+1 (depending on parity) where β* is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K1's and K2's. Similar results are proven for bipartite graphs.",

author = "Golumbic, {Martin Charles} and Laskar, {Renu C.}",

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N2 - A set of vertices X is called irredundant if for every x in X the closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 2β* or 2β*+1 (depending on parity) where β* is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K1's and K2's. Similar results are proven for bipartite graphs.

AB - A set of vertices X is called irredundant if for every x in X the closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 2β* or 2β*+1 (depending on parity) where β* is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K1's and K2's. Similar results are proven for bipartite graphs.

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Irredundancy in circular arc graphs

Abstract

Bibliographical note

ASJC Scopus subject areas

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