Toughness and Matching Extension in {\mathcal{P}_3}-Dominated Graphs |
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Authors: | Metrose Metsidik Elkin Vumar |
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Institution: | 1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China 2. College of Mathematics, Information and Physics Sciences, Xinjiang Normal University, Urumqi, 830054, People’s Republic of China
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Abstract: | Let G be a connected graph. For ${x,y\in V(G)}$ with d(x, y) = 2, we define ${J(x,y)= \{u \in N(x)\cap N(y)\mid Nu] \subseteq Nx] \,{\cup}\,Ny] \}}$ and ${J'(x,y)= \{u \in N(x) \cap N(y)\,{\mid}\,{\rm if}\ v \in N(u){\setminus}(Nx] \,{\cup}\, Ny])\ {\rm then}\ Nx] \,{\cup}\, Ny]\,{\cup}\,Nu]{\setminus}\{x,y\}\subseteq Nv]\}}$ . A graph G is quasi-claw-free if ${J(x,y) \not= \emptyset}$ for each pair (x, y) of vertices at distance 2 in G. Broersma and Vumar (in Math Meth Oper Res. doi:10.1007/s00186-008-0260-7) introduced ${\mathcal{P}_{3}}$ -dominated graphs defined as ${J(x,y)\,{\cup}\, J'(x,y)\not= \emptyset}$ for each ${x,y \in V(G)}$ with d(x, y) = 2. This class properly contains that of quasi-claw-free graphs, and hence that of claw-free graphs. In this note, we prove that a 2-connected ${\mathcal{P}_3}$ -dominated graph is 1-tough, with two exceptions: K 2,3 and K 1,1,3, and prove that every even connected ${\mathcal{P}_3}$ -dominated graph ${G\ncong K_{1,3}}$ has a perfect matching. Moreover, we show that every even (2p + 1)-connected ${\mathcal{P}_3}$ -dominated graph is p-extendable. This result follows from a stronger result concerning factor-criticality of ${\mathcal{P}_3}$ -dominated graphs. |
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