Anti-forcing numbers of perfect matchings of graphs

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• 作者：Hongchuan Lei^a ; ^b ; Yeong-Nan Yeh^b ; ^{mayeh@math.sinica.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Heping Zhang^c ; ^{zhanghp@lzu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Plane bipartite graph ; Hexagonal system ; Perfect matching ; Forcing number ; Anti-forcing number ; Fries number
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 March 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：95-105
• 全文大小：476 K

文摘

We define the anti-forcing number of a perfect matching class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si1.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=dbff034d01584dc47d3fc341e590df84" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$M$ of a graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si16.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=a7a5f992a194c5ec29fe204a7776299c" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ as the minimal number of edges of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si16.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=a7a5f992a194c5ec29fe204a7776299c" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ whose deletion results in a subgraph with a unique perfect matching class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si1.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=dbff034d01584dc47d3fc341e590df84" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$M$, denoted by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si19.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=91b4f16174dae78348d01d24193391df" title="Click to view the MathML source">af(G,M)class="mathContainer hidden">class="mathCode">$af(G,M)$. The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si16.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=a7a5f992a194c5ec29fe204a7776299c" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ with a perfect matching class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si1.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=dbff034d01584dc47d3fc341e590df84" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$M$, we obtain a minimax result: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si19.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=91b4f16174dae78348d01d24193391df" title="Click to view the MathML source">af(G,M)class="mathContainer hidden">class="mathCode">$af(G,M)$ equals the maximal number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si1.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=dbff034d01584dc47d3fc341e590df84" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$M$-alternating cycles of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si16.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=a7a5f992a194c5ec29fe204a7776299c" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ where any two either are disjoint or intersect only at edges in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si1.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=dbff034d01584dc47d3fc341e590df84" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$M$. For a hexagonal system class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si26.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=8e335cc633806dd6a00a586dc02b7364" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$, we show that the maximum anti-forcing number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si26.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=8e335cc633806dd6a00a586dc02b7364" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$ equals the Fries number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si26.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=8e335cc633806dd6a00a586dc02b7364" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$. As a consequence, we have that the Fries number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si26.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=8e335cc633806dd6a00a586dc02b7364" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$ is between the Clar number of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004394&_mathId=si26.gif&_user=111111111&_pii=S0166218X15004394&_rdoc=1&_issn=0166218X&md5=8e335cc633806dd6a00a586dc02b7364" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$ and twice. Further, some extremal graphs are discussed.