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zPT{TJMlJ_7h8IJeW{{J0-a9qczGmvaZL9nez#UTq!z1(hub6>{_6$@1%W3!Q!AG`+Go^W^-SJ=W7pR;nvS=?ZB_#l-NtK>)|%0YRNq;X@0~ao zQ(|<~^D|hNY391e*TKc5q2cKl7we&;1}3>t>mk~Wz{-+ey_tRJ;>q>y+4VW=V@kq| zijA9o|A@(hBv6x94M^8A>BxPP&5GA1t@r;n?Qh+_o9d9DHn2ZEqjuIIuXbg~D2(Z? zwHNYQ%e1%5{tq@i(QWrm^{%?gm^9mKbIN{$%MJrbO`o{^TJ#E#A_MbBudcRD*j%k& zc>Cv)h%!iGGrD$X?ExRK8x7I|cWsS6|2C)Z)0g + +startlet Gundirectedgraphlet kpositiveintegerminimum maximal matchingon G with at most kedgesthere existsmatching in Gthere exists nomatching in Gthere are two outcomes of runningthe algorithm: there is such amatching, or there is no suchmatching.we must prove that this decisionproblem has a polynomial kernel,which would imply that it is fixed-parameter tractable.black-box proof: let M be a maximal matching in G. (1) if |M| > k, then k is too small => no such matching found (2) if |M| <= k, then |M| is within budget let V(M) denote the vertices covered by M similar to vertex cover problem it follows that V(M) <= 2k, since each edge in M covers at most two vertices let u vertex outside of V(M) then N(u) can only contain vertices in V(M), since if v in N(u) and v not in V(M) would mean M is not maximal, thus contradiction. this gives a reduction rule: remove all vertices outside of V(M) then form G[V(M)], the fully connected subgraph induced by V(M) => return G[V(M)] as kernel G[V(M)] is a polynomial kernel because - the number of vertices is at most 2k - the number of edges is 2k(2k-1)/2 = 2k^2 - k (since fully connected, undirected) - the kernel size is thus O(k^2), polynomial in k Q.E.D. \ No newline at end of file