1976
DOI: 10.1016/s0095-8956(76)80005-6
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The strong perfect-graph conjecture is true for K1,3-free graphs

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Cited by 119 publications
(24 citation statements)
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“…Let G be a claw-free graph. If (G) ≥ 3, then by Lemma 2 and the Strong Perfect Graph Theorem [4] (which was proved in the special case of claw-free graphs by Parthasarathy and Ravindra [21]), we know G is perfect if and only if it contains no odd hole. Thus, in order to know if G contains an odd hole we just have to check if it is perfect, which can be done in polynomial time [6].…”
Section: Theorem 7 If G Is a Claw-free Graph Then It Is Hereditary mentioning
confidence: 96%
“…Let G be a claw-free graph. If (G) ≥ 3, then by Lemma 2 and the Strong Perfect Graph Theorem [4] (which was proved in the special case of claw-free graphs by Parthasarathy and Ravindra [21]), we know G is perfect if and only if it contains no odd hole. Thus, in order to know if G contains an odd hole we just have to check if it is perfect, which can be done in polynomial time [6].…”
Section: Theorem 7 If G Is a Claw-free Graph Then It Is Hereditary mentioning
confidence: 96%
“…Let e = xy and M − e be a monster. By [1], M is K 1,3 -free but M − e is not since M + e is a monster as well by Lovász's Perfect Graph Theorem [13], and the SPGC is true for K 1,3 -free graphs by [17]. Let us now consider a K 1,3 = {w, x, z; y} in M − e, i.e., a K 1,3 with center y in M + e. The nodes w, y, z and x, y, z, resp., induce P 3 's in M + e, which are contained in the holes C w and C x , resp.…”
Section: Case 2 E Is H-critical Let E = Vmentioning
confidence: 99%
“…Thus, x and y form an H-pair in F . [17]. Let us now consider a K 1,3 = {w, x, y; z} with center z in M − e. The nodes x, z, w and y, z, w, resp., induce P 3 's, which are contained in the holes C x and C y , resp.…”
Section: Theorem 1 Let G Be a Perfect Graph And The Linegraph Of A Gmentioning
confidence: 99%
“…Claw-free Berge graphs have been shown to be perfect by Parthasarathy and Ravindra [101] by exploiting properties of minimally imperfect graphs. Another proof, based on properties of minimally imperfect graphs, is given by Giles, Trotter and Tucker in [75].…”
Section: Claw-free Graphsmentioning
confidence: 99%